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The LyX User's Guide - Lund University Publications

1. Table 18 Parameter values estimated with IFM method for observations in DK2 Price residuals are assumed t distributed while wind power residuals are skewed normal 70 G DK1 Extreme dependence Below is the estimated parameters of the marginal functions estimated using the full maximimum likelihood FML method and the inference functions for margins IFM method Both block maxima of price residuals and wind power residuals are assumed to be GEV distributed DK Marginal function Price Marginal function WP Location Scale Shape Location Scale Shape Gumbel 1 362 0 082 0 543 0 069 0 324 0 091 1 448 0 053 0 346 0 037 0 193 0 093 Tawn 1 372 0 089 0 567 0 083 0 351 0 105 1 428 0 052 0 329 0 034 0 128 0 103 Husler Reiss 1 362 0 081 0 543 0 068 0 324 0 096 1 448 0 053 0 346 0 037 0 193 0 097 Galambos 1 358 0 082 0 545 0 069 0 347 0 096 1 449 0 053 0 346 0 037 0 191 0 096 Table 19 Estimated parameter for marginal functions FML The block max ima observations are from DK1 Parameter estimates standard variations are given in parentheses Marginal functions ee Location Scale Shape Price residual BM 1 3621 0 5425 0 3240 WP residual BM 1 4484 0 3458 0 1927 Table 20 Parameter values of marginal functions estimated wi
2. fo o s a O O fo oO o oa O o9 oF O o wo 2 Q O a T T T T T T T T T T T T 0 0 0 2 0 4 0 6 0 8 1 0 0 0 0 2 0 4 0 6 0 8 1 0 Quantile Quantile Figure 24 y plot and plot for DK2 Dashed lines indicates the 95 confi dence interval The x u function value is greater than zero for all u indicating a positive relationship between the price residual and the additive inverse wind power residual However the x u is smaller than 1 for u 1 which supports the idea of asymptotic independence Chi Plot Chi Bar Plot SP el e eee aj qe wN q Sz wo oO Oo 2 a o O 9S 9 T wo wo Si D SS z T T a T T T T T T 0 0 1 0 0 0 0 2 0 4 0 6 0 8 1 0 Quantile Quantile Figure 25 y plot and y plot for price and WP residual block maxima in DK2 Dashed lines indicates the 95 confidence interval Due to large variance of the functions conclusions regarding dependence and asymptotic dependence are difficult to reach 59 3 5 4 Discussion on extreme value dependency The results of the copula and marginal function fitting for the block maxima indicate that no significant dependence between the maximum price residual and maximum additive inverse of WP residuals exists in neither DK1 nor DK2 In other words the extreme excess increases in price appear independent from the extreme excess decreases in the produced WP The conclusion is supported with the estimated dependence parameter
3. Chi Plot Chi Bar Plot o J gt J W W o 7 o 7 zo o O E O W N o 97 gt o ae T T T T T j 0 0 0 2 0 4 0 6 Quantile T 0 8 T 1 0 T 0 0 0 2 0 4 0 6 0 8 1 0 Quantile Figure 22 y plot and plot for DK1 Dashed lines indicates the 95 con fidence interval The x u function value is greater zero for all u indicating a clear and positive relationship between the price residual and the additive inverse wind power residual However the x u is smaller than 1 for u 1 which means asymptotic independence Chi Plot Chi Bar Plot o all sess oe a zA ogi Nis E Vy P ane wi iJ Kat ey re re O O m oO o ao E a O o T o O as i 1 r y s M wah a A 4 tok a ve Mar SM hy a F wR tu Ws te 9 i nee E a yak a 1 SE E ot Bre 1 o WoW o k st 0 0 0 2 0 4 0 6 0 8 1 0 Quantile Quantile Figure 23 y plot and x plot for price and WP residual block maxima in DK1 Dashed lines indicates the 95 confidence interval x u is close to 0 for all values of u indicating independence between the price residual and the additive inverse wind power residual Likewise u is is different from 1 for u 1 which means asymptotic independence Large estimate variation impede reaching conclusions based on the plots however no clear sign of dependence nor asymptotic dependence exists 58 Chi Plot Chi Bar Plot
4. ACF Z l 0 02 0 06 i Lag days d ACF of the squared price residuals in DK2 resulting from fitting ARMA 4 3 GARCH 1 1 model with weekday indicator variables to the first difference price series The dashed line indicates the threshold for sig nificance at a 95 confidence level There is no indication of significant autocorrelation as no ACF values surpass the threshold and the residuals can be assumed homoscedastic Price filtering Q DK1 First difference daily Wind Power ki a 5 R4 3 a 2 D E ne t o eo 99 oO 4 T T T T 2012 2013 2014 2015 t year a First difference of daily wind power pro duced in DK1 The series is assumed station ary as the KPSS test can not reject the station arity hypothesis on a 95 confidence level DK1 ACF First difference daily WP 0 05 ACF 0 00 0 05 0 10 Lag days c Sample ACF of the daily wind power resid ual in DK1 No significant autocorrelation can be found in the residuals as no ACF values surpass the dashed lines which indicates the 95 significance level DK1 Wind Power Filtering DK1 Wind power residual rae a 4 T 2 N 2 Z Da 4 T T T T 2012 2013 2014 2015 t year b Daily wind power residual in DK1 DK2 ACF Squared WP residuals O gy _ passes sees secesnesteessesesssosseeeedaconees oO welll Li
5. Likewise the ACF of the squared first difference WP gives significant val ues for some lags indicating that heteroscedastic effects should be taken into account when modeling the series Model selection The ARMA 3 3 GARCH 2 3 model is selected according to the AIC and the parameter estimates are presented in Table 9 The resulting residuals are assumed to be normally distributed The ACF of the residuals shows no sign of significant autocorrelation Like wise the weighed Ljung Box tests of the residuals do not reject the assump tion of no autocorrelation The Ljung Box tests of the squared residuals can also not reject the hypothesis of no ARCH effects on a 95 confidence level and together with no significant ACF values of the squared residuals it is con cluded that no further significant heteroscedasticity exists in the WP residuals Ljung Box test results are presented in Table 10 and the ACFs can be found in Appendix D together with residual plot 34 DK2 ACF First difference daily WP 0 05 0 05 ACF 0 15 0 25 Lag days ACF 0 05 0 00 0 05 010 0 15 0 15 0 10 DK2 ACF Squared first diff daily WP Lag days Figure 8 Left ACF of the first difference of daily wind power produced in DK2 The dashed lines indicate significance threshold for a 95 confidence level Right ACF of the squared first difference of daily wind power produced in DK2 Signific
6. of filtering data with an ARMA model After fitting the ARMA GARCH model the standardized residuals are checked for autocorrelation with visual inspection and with the application of the Weighted Ljung Box test to the residuals The magnitude of the autocorrelation on individual lags can be inspected by looking at the sample autocorrelation function ACF of the residuals Values of the sample functions departing greatly from zero indicate autocorrelation at the given lag and may lead to rejection of the model for not successfully resulting in autocorrelation free residuals The limits for significance at a 95 level is set as Xo 975 n where n is the sample size and Ag 975 is the 97 5 upper quantile of the standard normal distribution Hence the estimated ACF values at individual lags has to be greater than Ao 975 n or smaller than Ao 975 n in order to be considered significantly different from zero Moreover in order to take into account the autocorrelation magnitudes as group the Weighted Ljung Box test suggested by Fisher and Gallagher 2012 is applied In the Weighted Ljung Box test applied to residuals from an ARMA p q model the test statistic is given as K p rr 2 te 4 where n is the sample size and the estimated autocorrelation of lag k fk and the weight w k are given with the following equations 1 n k Pr 2 Et E E E K k 1 ti ae Under the null hypothesis that the residuals
7. 7 o faa m o o o kag O o 9 o o o z o oe n J o o o O o o o o o ce e o 2 Jo 2 S o ony T T T T T oT T T T T 0 0 0 2 0 4 0 6 0 8 1 0 0 0 0 2 0 4 0 6 0 8 1 0 u BM price residual u BM price residual Figure 16 Non parametric uniform transform of the block maxima price resid ual vs wind power residual in DK1 and DK2 respectively For DK1 a 20 days block length is chosen while the block length is set to 40 days for DK2 In Figure 15 the componentwise block maxima BM of price residual vs additive inverse of WP residuals in DK1 and DK2 are plotted The corre sponding scatterplots of the non parametric uniform transform are shown in Figure 16 3 5 1 DK1 Extreme dependence in West Denmark The block length is chosen by investigating the QQ plots of GEV distributions fitted to block maxima of different block lengths In Figure 17 the QQ plot for block maxima price residuals as well as block maxima WP residuals are shown for block length of 10 20 30 and 40 days Larger block lengths than 40 days will result in very few observations thus are not considered The block maxima of WP residuals seem to follow GEV distribution for block length 20 and the price residual block maxima follow a GEV distribution for any choice of block length with the exception of maximum observation of price residuals which is much higher than expected under the GEV assumption This applies for any of the chosen block lengths Hence the
8. MA b2 2 300945 03 0 763197 CG 48 62527 eS 13 41433 C3 5 269275 Weekday parameters C4 3 900434 Cs 9 363129 Ce 30 08975 Cr 13 55165 wW 75 1294 GARCH a 0 172647 B 0 816063 t dist shape parameter v 4 266136 Table 5 Parameter estimates for the model for daily power prices in DK2 The chosen model is ARMA 4 3 GARCH 1 1 with weekday indicator variables and student t distributed residuals Ljung Box test Test statistic p value Residuals lag 20 weighed 6 0431 1 Residuals lag 30 weighed 9 901 0 9913 Residuals lag 50 weighed 17 2826 0 9775 Squared residuals lag 12 2 3364 0 9987 Squared residuals lag 20 4 7286 0 9998 Squared residuals lag 30 8 1356 1 Table 6 Weighed Ljung Box test for residuals and classic Ljung Box test squared residuals resulting from modeling the electricity prices of DK2 with ARMA 4 3 GARCH 1 1 In case p values are smaller than 0 05 the null hy pothesis of no autocorrelation in the residuals and squared residuals respec tively can be rejected at a 95 confidence level As all p values are greater than 0 05 the null hypothesis can not be rejected thus no significant autocor relation nor heteroscedastic effects can be identified in the residuals 30 The ACF of the residuals indicates no remaining autocorrelation and this is supported by the weighed Ljung Box test presented in Table 6 which cannot reject the assumption o
9. The dashed lines in the plots indicate the threshold for the function values to be significant at a 95 level DK1 ACF First difference daily price DK1 ACF Squared first diff daily price pai _ S O Lolu zd 3 T L I l l T Tt TT o lt a a O 2 oJ o W o ga 2g s a 5 Q ke ao Q a EEE a rr Re E 5 4 T T T T T T T T T T T T T 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag days Lag days Figure 3 Left Sample ACF of first difference of daily power prices in DK1 Negative autocorrelation for lag 1 is the result of mean reversion Peaks at lag 7 14 21 and 28 support the idea of weekly periodicity in the price series The dashed lines indicate the threshold for 95 significance level Right Sam ple ACF of squared first difference of daily power prices in DK1 Significant value for lag 1 indicates heteroscedasticity and justifies the use of GARCH specification in the model 26 As expected for mean reverting processes the autocorrelation with lag 1 of the first difference series is negative Mean reversion secures that the prices return to the mean level which implies that large prices tend to be followed by lower prices and low prices tend to be followed by higher A clear weekly periodicity exists expressed as ACF peaks for lags that are a multiples of seven This support the choice of using the weekday indicators in the model The ACF of the squared first difference
10. can be defined as _ __alog Pr U gt u _ _ 2Zlog 1 u x u log Pr U gt u V gt u s5 logC u u 1 forO lt u lt l and X limy 1V w With this definition u lies between 1 and 1 and asymptotic dependence results in y 1 For independent variables C u v 1 u 1 v so that x u 0 for all ue 0 1 By plotting x u against u the dependence can be inspected It can be proved that for Normal copula the y is equal to the dependence parameter p and the convergence towards p is approximately linear for u gt 0 5 This makes the inspection of the tail dependence easier in the y plot compared with the y plot 24 3 Analysis and results 3 1 Data The data used in this thesis consist of 1080 observations of the daily day ahead electricity prices and the daily wind power produced in the two areas DK1 and DK2 for the period from 1 January 2012 to 15 December 2014 Prices are given as daily average in Danish Krone DKK and the daily wind power is the aggregated hourly production measured in MWh Both series are publicly available on the website of Nord Pool Spot 3 2 Daily prices The daily power prices are presented in Figure 2 for the areas of DK1 and DK2 The average price of 1 MWh during the period was DKK 264 47 and DKK 272 21 for the areas DK1 and DK2 respectively On 7 June 2013 the price reached DKK 3253 in DK1 which according to Energinet dk Rasmussen 2013 was the result of reduc
11. 4518 0 3489 0 7131 0 02847 0 5 Husler Reiss 0 8964 0 2683 1 113 0 04545 0 6968 Galambos 0 5065 0 2114 1 062 0 05544 0 6299 Gumbel 1 176 0 1565 0 8943 0 04046 0 5829 z Tawn 0 3415 0 3570 0 3912 0 02348 0 525 Husler Reiss 0 8063 0 2258 1 199 0 05544 0 6998 Galambos 0 4371 0 1760 1 114 0 04046 0 6119 Table 14 Goodness of fit test results for extreme value copulas fitted to BM of DK1 using FML CML and IFM None of the models can be rejected on a 95 confidence level by the M test as all p values are greater than 0 05 However the S test produces low p values for all copulas and rejects Gumbel and Tawn copula for all estimation methods Full maximum likelihood Dependence parameters estimated by FML do not clearly differ from the parameter values leading to independence None of the copula models can be rejected on a 95 confidence level according to the goodness of fit test for extreme value copulas with test statistic M how ever the S test rejects Gumbel and Tawn copula Test results are presented together with estimated copula parameter values in Table 14 Estimated pa rameter of the GEV distributions fitted to margins can be found in Appendix H and the results show that the parameters of the marginal distribution func tion agree throughout all four copula models Canonical maximum likelihood The results of the CML fitting of ext
12. On 1 835061 AR og 1 357896 3 0 4471407 0 2 136831 MA b2 1 614041 03 0 4704461 Gy 53 81275 Cg 6 287118 C3 3 096599 Weekday parameters C4 4 074114 Cs 9 863092 Ce 29 92216 C7 18 68064 w 1115 457 GARCH a 0 3651083 B 0 2925828 t dist shape parameter v 3 662627 Table 3 Parameter estimates for the ARMA 3 3 GARCH 1 1 model with weekday indicator variables and student t distributed residuals for first differ ence of daily electricity prices in DK1 Ljung Box test Test statistic p value Residuals lag 20 weighed 6 4077 0 9989 Residuals lag 30 weighed 10 0118 0 9792 Residuals lag 50 weighed 14 9243 0 9952 Squared residuals lag 12 0 4522 1 Squared residuals lag 20 0 4809 1 Squared residuals lag 30 0 5556 1 Table 4 Weighed Ljung Box test for residuals and classic Ljung Box test for squared residuals resulting from modeling electricity prices in DK1 with ARMA 3 3 GARCH 1 1 model Null hypothesis of no autocorrelation is re jected for p values smaller than 0 05 The null hypothesis cannot be rejected for any of the tests 28 DK2 ACF First difference daily price DK2 ACF Squared first diff daily price e G4 pe ee ee ee g Ste dtl an 9 d T T T T T T T T T T T T T T 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag days Lag days Figure 4 Left ACF of first difference of power prices in DK2 There ex
13. and if yes in which way e Does unusual low wind power in feed to the electricity system increase the risk of extremely high prices What could be the explanatory factors for this result Inferences on the relation between the wind power in the electricity system and the prices will be based on extreme value and copula theory The research conducted in order to answer the questions above is divided into four steps 1 Filtering electricity prices in order to remove predictable weekly effects as well as general trends 2 Filtering wind power production in order to remove predictable effects and autocorrelation from observations 3 Investigating the general dependence structure of the full sample of fil tered observations using copulas 4 Investigating the dependence between the maximum daily prices and minimum daily wind power production in the filtered sample using ex treme value copulas 1 1 Previous studies Several researches point at relationship between the share of wind power and prices The findings of Lindstr m and Regland 2012 indicate that markets with high share of renewable energy have higher probability of having spikes Ketterer 2014 shows that increased wind power in the German electricity system overall reduces electricity price level but increase its volatility Study ing the impact of photovoltaic and wind energy on prices of power traded at the European Power Exchange Paraschiv et al 2014 l
14. block length is set to 20 days as neither shorter nor longer block lengths offer improvements to the fit 48 Price residual BM WP residual BM 10 10 pa o 2 2 2 Zz oS o T S S e og e a 2 wo o 5 5 9S 1 223 4 5 05 10 15 20 25 Model Quantiles Model Quantiles Price residual BM WP residual BM 20 20 pal o 2 p 2 2 8 2 5 o N o 2 fE 3 ao en 2 2 a ao 5 oS T 2S SD SS 1 0 15 2 0 Model Quantiles Model Quantiles Price residual BM WP residual BM 30 30 wo val 2 aS EOS E o 6 aN o a o2 2 Qe a n Ee ES T2 3 4 6 12 16 20 24 Model Quantiles Model Quantiles Price residual BM WP residual BM 40 40 o o 2 i E S oa oa N i F GS ao x a Be O O E a a E E w w oe te SS 1 4 1 8 2 2 Model Quantiles Model Quantiles Figure 17 QQ plots for price residuals and wind power residuals block maxima BM in DK1 for block lengths 10 20 30 and 40 While the maximum price residual seems to lie outside the GEV distribution for all of the plotted block lengths the block maxima wind power residuals follow the GEV already for a block length of 20 49 DK1 Density of price res BM DK1 Density of WP res BM oH bad b Enmpirica Empiric jan 2 j Modeled Modeled 37 cs oO Gs 2 a 24 a Nn O h nj o oO 2 Dees E E E a ko JN o gt T T T T T T T T T T T 0 5 10 15 20 25 0 5 1 0 15 2 0 25 N
15. dependence structure prices into first mod eling the time series and secondly fitting copulas to the filtered series 2 1 Price model The electricity prices vary over the day week and year due to the varying de mand of electricity In order to make conclusions on the relationship between the price and the wind power production it is necessary to remove predictable time varying effects in other words create a series of independent identically distributed IID price residuals as the result of filtering the prices Due to the fairly complicated structure of the electricity prices including mean reversion seasonality spikes and volatility clustering it is not easy to find an appropriate model for the electricity price to be used as filter However academic litera ture offers many suggestions such as hidden markov chain models state space models and models using external regressors In this section the model used in this thesis will be presented and reasons behind the model will be explained The notation regarding the model is used as in Cryer and Chan 2008 and is in the following explained Return measures not used Given a series of prices P the return is de fined as R ee and the continuously compounded return log return as r log P log P _ 1 In classic financial modeling it is common to use re 8 turns or log returns instead of prices as the return series tend to be stationary while price series are general
16. from DK1 are presented in Table 11 In this table goodness of fit test results are also shown for each of the copulas and each estimation method Marginal function parameters estimated with the FML and IFM methods are found in Appendix E Log Goodness of fit DK1 Copula Parameter likelihood ca Palle Normal 0 6359 0 018 2627 334 0 0279 0 08142 Clayton 1 106 0 076 2681 599 0 4784 0 0004995 Gumbel 1 73 0 05 2659 486 0 1745 0 0004995 ce Frank 4 878 0 234 2636 792 0 0237 0 1204 0 6365 0 018 t 49 72 38 03 2627 74 0 0286 0 07343 Normal 0 63800 0 01526 278 7 0 0279 0 07842 Clayton 1 06044 0 06044 220 3 0 4784 0 0004995 Gumbel 1 67058 0 04017 239 3 0 1745 0 0004995 O Frank 4 8363 0 2169 269 3 0 0237 0 1334 0 63827 0 01544 t 102 652 137 09 278 6 0 0281 0 06643 Normal 0 6344 0 0154 278 5 0 0279 0 09441 Clayton 1 0337 0 0592 219 2 0 4784 0 00049 Gumbel 1 6664 0 0400 238 3 0 1745 0 00049 Frank 4 8213 0 2163 269 8 0 0237 0 1124 0 6347 0 0155 t 103 39 137 80 278 4 0 0286 0 08641 Table 11 DK1 Dependence parameter estimate and goodness of fit for cop ulas when using FML CML and IFM Taken the estimator variance into consideration all parameter values agree across estimation method The cop ula models are rejected for p values smaller than 0 05 The Frank copu
17. modeling the distribution of one observations is not possible In order to approximate the GEV distribution the data is divided into blocks and the modeling of the GEV marginal distribution functions and the extreme value copulas will be based on the block maxima With this method the bi variate data series X Y is divided into blocks of length l and for each block the component wise maxima are found In other words for the blocks ie 1 2 N of length l the component wise block maxima is given by Mi Mix Miy 13 For appropriate choice of block length the maxima will be approximately GEV distributed thus GEV distributions are fitted to the series M i x and M iy According to Coles 2001 selecting a block length is ultimately a question of trade off between bias and variance Long blocks will result in few observations resulting in large variance of the parameter estimates while short block lengths will give more observations but not from the GEV distribution as the GEV by definition is the limit distribution of M for n gt oo The series of component wise block maxima M may not consist of ob served pairs from the original series however it is still useful for inferences on the joint extreme value behavior 2 3 3 Estimation method The copula parameters are estimated using three difference methods Full Max imum Likelihood FML Canonical Maximum Likelihood CML and Infer ence Functions for Margins IFM Th
18. of daily prices also presented in Figure 3 are significant for lag 1 thus volatility clustering should be taken into account in the model This is done through modeling the conditional variance within the GARCH framework Model order selection According to the AIC the ARMA 3 3 GARCH 1 1 model with weekday indicators and student t distributed residuals is selected and its estimated parameter values are presented in Table 3 The sample ACF of the residuals shows no significant values which indi cates that all autocorrelation in the price series has successfully been filtered away with the selected model In particular the weekly periodicity has been been removed as no clear autocorrelation on lags that are multiples of seven ex ists The test results of the Ljung Box tests are presented in Table 4 Weighed Ljung Box tests for the residuals result in high p values thus support that no further autocorrelation can be detected Moreover the sample ACF of the squared residuals is not significantly dif ferent from zero for any non zero lags Together with the test results of the Ljung Box test for squared residuals in which null hypothesis of no autocorre lation can be rejected for p values smaller than 0 05 it can be concluded that there exist no further significant heteroscedastic effects ACFs and residual plot can be found in Appendix A 27 Parameter Estimated value
19. significant for any non zero lag Likewise the Weighed Ljung Box test cannot reject the hypothesis of no autocorrelation on a 95 confidence level as no p values are smaller than 0 05 The Ljung Box test for the squared residuals can also not reject lack of heteroscedastic effects Test results are presented in Table 8 33 Ljung Box test Test statistic p value Residuals lag 20 weighed 5 2545 0 9846 Residuals lag 30 weighed 8 8678 0 9748 Residuals lag 50 weighed 17 2072 0 9551 Squared residuals lag 12 6 7367 0 8745 Squared residuals lag 20 15 75 0 732 Squared residuals lag 30 21 749 0 863 Table 8 Weighed and classic Ljung Box test for wind power residuals and squared residuals respectively With no p values smaller than 0 05 the null hypothesis of no autocorrelation is accepted at a 95 confidence level for any of the chosen lags Observations are from DK1 The ACF of the squared residuals is non significant for every non zero lag thus the residuals can be assumed homoscedastic ACFs and residual plot can be found in Appendix C 3 3 2 DK2 Wind power production in East Denmark The series of first difference WP in DK2 is assumed stationary as this assump tion cannot be rejected at a 95 confidence level by the KPSS test Clear autocorrelation of short lags is seen in the ACF of the first difference WP presented in Figure 8 indicating that ARMA model is appropriate
20. skewed 37 3 4 Dependence structure and copula fitting After filtering the series of daily prices and wind power resulting in IID resid uals the dependence structures between the residuals can be investigated Figure 11 contains plots of the price residuals vs the additive inverse of WP residuals The additive inverse is used instead of the WP residual itself as it is expected that low wind power is associated with high prices thus after changing the sign of the WP residuals a positive correlation linear relation ship to the price residuals can be found When using additive inverse of the WP residuals the low values of the WP residuals associated with large values of the price residuals will be found in the upper right corner of the scatter plots The positive relation is preferred when examining the dependence in particular in relation to the extreme dependence as the relation between the minimum WP residual and the maximum price residual can be modeled with the methodology of bivariate maxima introduced in Section 3 5 As expected the scatterplots in Figure 11 indicate a positive correlation the price residu als and the additive inverse of the WP residuals in DK1 as well as in DK2 This relationship will be investigated further in the following sections in which bivariate copulas will be fitted to the data The five copulas Normal Clayton Gumbel Frank and t copula presented in Table 1 are fitted to the full sample of the p
21. y plot Independence between bivariate data can be identified using x and y plots The plots are based on the definition of the asymptotic upper tail dependence as in equation 12 and idea of the y plot is in the following explained as in Coles et al 1999 The upper tail dependence is given as x limus P U gt u V gt u Observing that P U gt u V gt u gt 2 sgt Cee foru gt 1 and C u u Pr U lt u V lt u the chi function x u is be defined as _ log PrU lt u V lt u logu xlu 2 0 lt u lt l1 20 By this definition of the y functions the upper tail dependence is given as X limu x u For independent variables U V the x u is 0 for any permitted value of u If U V are upper tail independent x u 0 for u gt 0 Plotting x u against u gives the basis for determining overall lack of dependence and in the data In case the data is dependent but asymptotic independent y u may con 23 verge very slowly towards zero for u 1 resulting in y u being significantly greater than zero even for u values close to one This makes it difficult to identify the upper tail independence thus a new dependency measure X is introduced Denoting the survivor function Pr X gt 2 Y gt y by F x y the copula C can be defined as F x y 1 Fx 2 Fy y F x y C Fx 2 Fy y where C u v 1 u v C u v Similarly to the definition of y u in equation 20 x u
22. 1 as well as in DK2 indicate a significant dependence between the price residual and WP residuals Moreover for DK1 the Frank copula has the highest likelihood and highest goodness of fit p value for most of the used estimation methods The Normal copula gave the best goodness of fit for the data of DK2 The degrees of freedom of the t copula is very large for both DK1 and DK2 regardless of estimation method and as the t copula 45 converges toward the Normal copula as the degrees of freedom approaches in finity the fitted t copulas are very similar to the Normal copula This explains the large goodness of fit of the t copula as well as the dependence parameter of the t copula being very close to the dependence parameter of the normal copulas Since the first difference is used for modeling the prices as well as the WP the residuals express an excess change in price and WP The significant depen dence between price residual and WP residual can accordingly be interpreted as excess increase in wind power production leads to an excess decrease of the price while excess decrease in wind power production leads to excess in crease of the price This result is expected Wind power has low marginal cost compared to other kinds of power generation methods and high wind power production should therefore lead to low electricity prices while low production should lead to higher electricity prices The dependence parameter of the Normal copula which i
23. 165 0 2076 213 6 0 0271 0 06943 0 5798 0 0177 t 101 48 133 73 221 5 0 0256 0 1264 Table 12 DK2 Dependence parameter estimates and goodness of fit for cop ulas found with the three methods FML CML and IFM Taken the estima tor variance into consideration all parameter values agree across estimation method The Normal copula gives the best goodness of fit Neither Normal Frank and t copula can be rejected on a 95 confidence level Full maximum likelihood All dependence parameters estimated with FML are far from the parameter values leading to independence and the non zero dependence parameter of the normal copula indicates significant correlation between the variables The Normal Frank and t copula can not be rejected by the goodness of fit test at a 95 confidence level The Normal copula gives the highest log likelihood and the best goodness of fit The parameters of the marginal distribution functions are consistent across choice of copula model taking the variance of the estimates into considerations Canonical maximum likelihood The canonical maximum likelihood gives parameter estimates as in Table 12 which also shows the goodness of fit test results The dependence parameters are similar to the ones estimated with 44 FML Again Gumbel and Clayton copula can be rejected at a 95 confidence level while Normal Frank and t copula cannot be rejected according to the goodness of fit test Inferen
24. 29 DK2 Wind power filtering 68 E DK1 Full sample dependence Below is the estimated parameters of the marginal functions estimated using the full maximimum likelihood FML method and the inference functions for margins IFM method Price residuals are assumed t distributed while wind power residuals follow the skewed normal distribution DK Marginal function Price Marginal function WP Mean SD DF Mean SD Skew Normal 0 0098 0 024 0 988 0 053 3 659 0 421 0 0053 0 029 0 996 0 022 0 720 0 030 Clayton 0 0505 0 024 1 105 0 087 3 071 0 319 0 0165 0 029 1 02 0 021 0 649 0 028 Gumbel 0 0059 0 024 1 072 0 075 3 239 0 353 0 0088 0 029 1 01 0 022 0 803 0 033 Frank 0 0075 0 023 0 955 0 040 4 382 0 603 0 0110 0 029 1 01 0 023 0 692 0 031 t 0 0099 0 024 0 985 0 053 3 685 0 428 0 0052 0 029 0 997 0 022 0 721 0 030 Table 15 Parameters of marginal functions when using the FML Parame ters standard deviation is in parentheses Under all copula assumptions the estimated parameters of the marginal functions agree taking the estimate variation into consideration Marginal function price Marginal function WP Mean SD DF Mean SD Skew 0 01256 0 9566 4 175 0 009024 0 9989 0 6981 Table 16 Parameter val
25. 54 Bandwidth 0 2402 N 54 Bandwidth 0 1287 Figure 18 Density plots comparing the density of the fitted GEV distribution to the empirical density of the 20 day block maxima of price residuals and and WP residuals in DK1 The block length of 20 days result in 54 block maximum observations In Figure 18 the density function of the fitted GEV distributions are plotted and compared with the empirical density of the 20 day block maxima The plots suggest that the block maxima reasonably can be assumed to follow the GEV distributions Copula parameters estimated with the FML CML and IFM methods can be found in Table 13 together with goodness of fit test results Estimated GEV parameters of the margins can be found in Appendix G Full maximum likelihood Results of the FML estimation of GEV distri butions of margins and the various extreme value copulas indicate that depen dence parameters are very close to the parameter values leading to indepen dence None of the copula models can be rejected on a 95 confidence level according to both the goodness of fit test based on Sp and the goodness of fit test for extreme value copulas with test statistic M Moreover the parameters of the marginal GEV distributions agree through out all four copula models when taking estimator variance into consideration Canonical maximum likelihood CML fitting of extreme value copulas to observations in DK1 results in parameter estimates similar to those of t
26. P _1 Stationarity is an important feature of the series and it is needed for applying the filters The validity of the stationarity assumption will be examined by applying a unit root test In this thesis the test used to reject stationarity is the test of Kwiatkowski Phillips Schmidt Shin KPSS defined in Kwiatkowski et al 1992 ARMA In order to remove autocorrelation from the electricity price series an autoregressive moving average ARMA model for the first difference of prices is chosen The ARMA process which includes p AR terms and q MA terms is spec ified as p q AP amp 5 iA Pi bS jetij 1 i 1 j l Above AP is the first difference of the price at time t and e is the stochastic error term at time t The parameters and 0 relate to the AR and MA part of the model respectively and the model is referred to as ARMA p q Weekday indicator variables While a lot of electricity is consumed during the weekdays the electricity consumption is low during the weekend Due to this pattern in consumption the electricity prices follow a weekly pattern of low prices in the weekend and high prices during the weekdays In order to remove weekly effects from the resulting residuals weekday indicator variables are added to the ARMA p q model p q 7 AP e 5 PAR it gt Ojej 5 Cel ftew 2 i l j l k 1 1 ifte Ww Tew 0 otherwise Igew is the indicator variable activating the parameter when the day t fal
27. Wind power and its impact on power prices in Denmark Master thesis by Emilie Rosenlund Soysal May 2015 Abstract Including wind power in the European power system may cause in stability in the power market due to the uncontrolable generation capac ity The purpose of this thesis is to investigate the relationship between Danish power prices and wind power produced in Denmark both in terms of general dependence structure and the extreme dependence Filtered observations of daily power prices and filtered observations of daily wind power are modeled using copulas and extreme value the ory Results indicate a significant influence of the wind power on prices such that large wind power supply leads to low prices However it is shown that the extreme observations of high prices occur independently of those of low wind power leading to the conclusion that low wind power supply can not explain occurences of extremely high prices Keywords wind power power prices copula extreme value copula 1 Introduction The European energy supply is increasingly dependent on renewable energy sources In 2012 around 23 5 percent of the European Union s gross electric ity consumption originated from renewable energy sources however with the European Union s aim of a 50 percent share of renewable energy in the total energy consumption by 2050 an extensive transformation of the electricity production is required Wind energy is expected to pla
28. a Wh SLUTT Ea Oe e E E E E E ETE 4 T T T T T T T 0 5 10 15 20 25 30 Lag days d ACF of the squared wind power residual in DK1 The dashed lines indicate the signifi cance threshold at a 95 confidence level No significant autocorrelation can be seen in the plot indicating no further heteroscedastic ef fects exist in the residuals Figure 28 DK1 Wind power filtering 67 D DK2 Wind Power Filtering DK2 First difference daily Wind Power 20 1 10 WP first difference GWh 10 2012 2013 2014 2015 t year a First difference of daily wind power pro duced in DK2 DK2 ACF First difference daily WP 0 05 ACF 0 00 i 0 05 0 10 Lag days c Sample ACF of the daily wind power resid ual in DK2 The dashed line indicates the threshold of significance on a 95 confidence level No significant autocorrelation is identi fied as no ACF value for non zero lags surpass the threshold DK2 Wind power residual WP residual 2012 2013 2014 2015 t year b Daily wind power residuals in DK2 DK2 ACF Squared WP residuals 0 05 1 ACF 0 00 0 05 0 10 Lag days d Sample ACF of the squared daily wind power residuals in DK2 As no values surpass the 95 significance level threshold indicated with the dashed line homoscedasticity in the residuals can be assumed Figure
29. an be reduced to finding the marginal distributions Fx and Fy and the copula C In this thesis the Normal Gumbel Clayton Frank and t copula will be fitted to the residuals such that the best fitting copula can be selected The specifications of the five types of copulas to be fitted to the full sample of 15 Copulas Name C u v Parameter Independence Normal P u P v 1 lt p lt 1 p 0 Gumbel exp logu logv 1 lt 8 lt G 1 Clayton max u u3 1 0 1 lt 0 lt 0 0 6 0 Frank 4log 1 C lt 0 lt 0 0 0 0 t bys u t4 v v gt 0 1 lt p lt 1 p 0 v gt 0 Table 1 List of bivariate copulas used for modeling the dependence The table gives both the parametric description of the copula the permitted values of the parameters and the parameter values leading to independence is the inverse cumulative distribution function of the normal distribution is the bivariate normal distribution with linear covariance parameter p t is the inverse cumulative distribution function of the t distribution with v degrees of freedom while t is the bivariate cumulative t distribution residuals are presented Table 1 Note that the price residuals are denoted X and its corresponding uni form transform u while the wind power residuals are named Y with its the uniform transform v 2 3 2 Extreme value copu
30. ant values indicate the need for modeling the conditional variance Parameter Estimated value Q 0 4530423 AR Q2 0 01562591 os 0 1214103 0 0 01113139 MA b 0 5570294 03 0 3720104 W 1 011081 Qy 0 01173515 Q2 0 05301413 ADLN By 0 110688 Bo 6 51337e 11 b3 0 7784363 Table 9 Parameter estimates from fitting ARMA 3 3 GARCH 2 3 to the daily wind power production in DK2 39 Ljung Box test Test statistic p value Residuals lag 20 weighed 6 732 0 9973 Residuals lag 30 weighed 11 2434 0 9341 Residuals lag 50 weighed 17 6404 0 9648 Squared residuals lag 12 7 3896 0 8308 Squared residuals lag 20 11 7756 0 9236 Squared residuals lag 30 17 7639 0 9622 Table 10 Weighed Ljung Box test for residuals and classic Ljung Box test squared residuals resulting from modeling the daily WP of DK2 with ARMA 3 3 GARCH 2 3 As all p values are larger than 0 05 the assump tion of no autocorrelation hence homoscedasticity can not be rejected The QQ plots of the wind power residuals in DK1 and DK2 which com pare the empirical distribution of the residuals to the normal distribution are presented in Figure 9 The plots indicate that the residual distributions are slightly skewed and not symmetrical like the normal distribution The skew ness of the residual distributions is in conflict with the model assumption used fo
31. antiles from the student t distribu tion form a straight line Only the most extreme observations deviate from the line indicating that the student t tail distributions are thinner than that of the observations 31 DK1 Daily Wind Power Prod DK2 Daily Wind Power Prod 60000 80000 1 fi 10000 15000 20000 25000 Wind power MWh 40000 fi Wind power MWh 20000 i 5000 0 i 0 1 2012 2013 2014 2015 2012 2013 2014 2015 t year t year Figure 6 Daily wind power produced in DK1 and DK2 3 3 Wind power The daily wind power WP produced in DK1 and DK2 respectively is pre sented in Figure 6 The wind power has the lower bound zero and an upper bound equal to the installed capacity The maximum observed wind power generated during one day is approximately 80 GWh in DK1 and 25 GWh in DK2 The fitting of ARMA GARCH models to the wind power production is presented in this section and the choice of the model and the goodness of fit will be further discussed 3 3 1 DK1 Wind power in West Denmark In order to achieve stationarity the first difference of the WP is used for mod eling The KPSS test for stationarity indicates that stationarity cannot be rejected on a 95 confidence level for the first difference series The sample ACF is shown in Figure 7 Significant autocorrelation do exist for lag one and two suggesting that the ARMA specification for filtering the WP da
32. arket with its wind power share of more than 30 of the generated electricity does not experience wind power as the cause of extreme day ahead prices thus the large share of wind power is not challenging the stability of the day ahead market in terms of extremely high prices 62 References Beirlant J et al 2004 Statistics of Extremes Theory and Applications 1st ed Wiley Cap raa Faugeres and Genest 1997 A nonparametric estimation procedure for bivariate extreme value copulas Biometrika Volume 84 3 pages 567 577 Coles S 2001 An Introduction to Statistical Modeling of Extreme Values Springer Verlag London Coles S Heffernan J and Tawn J 1999 Dependence Measures for Extreme Value Analyses Extremes 1999 Volume 2 4 pages 339 365 Cryer J D and Chan K 2008 Time Series Analysis With Applications in R Springer Scienge Business Ea Energianalyse A S 2005 50 pct vindkraft i Danmark i 2025 en teknisk okonomisk analyse http energinet dk visited 2015 04 20 Fisher T J and Gallagher C M 2012 New Weighted Portmanteau Statistics for Time Series Goodness of Fit Testing Journal of the American Statistical Association Volume 107 498 Genest C et al 2011 A goodness of fit test for bivariate extreme value copulas Bernoulli Volume 17 1 pages 253 275 Genest C Quessy J and R millard B 2006 Goodness of fit Procedures for Copula Models Based on the Probability Integral Tran
33. ce functions for marginals All values resulting from the IFM es timation are consistent with the FML estimates As with the two other esti mation method all dependence parameters are far from those values leading to independence Normal Frank and t copula cannot be rejected 95 confidence level 3 4 3 Likelihood ratio test for Normal copula The likelihood ratio test is applied to the copula based on empirical marginal functions see the estimation results under CML method For DK1 the log likelihood of the normal copula is 278 73 and the log likelihood of independence copula is 2 418e 13 The resulting test statitics as defined in 19 is 557 46 and this is much greater than 3 84 which is the 95 upper quantile of the y 1 distribution The test rejects the null hypothesis of independence The log likelihood of the normal copula fitted to the empirical pseudo obser vations of DK2 is 222 33 while the log likelihood of the independence copula is 2 55le 13 This results in a test statistic with the value 444 67 witch is greater than the 95 upper quantile of the y 1 distribution thus the test rejects the hypothesis of independence The likelihood ratio test confirms the idea that there exists significant de pendence between the price residuals and the additive inverse of WP residuals 3 4 4 Discussion on full sample dependence The results of the copula and marginal function fitting for the price residuals and WP residuals in DK
34. clearing of financial contracts and the area prices differ from the system price only in case of the electrical congestion due to limited transmission capacity between the bidding areas Nord Pool Spot opens for intraday trading two hours after the closure of the day ahead market and the trading continues until one hour before delivery The intraday system Elbas gives market agents the opportunity to reach balance between the purchased quantities and expected delivered quantities Intraday balancing is needed in case the day ahead trades did not reach the agent s desired trade quantities or unpredicted incidents occur e g an offshore wind power plant produces less electricity than predicted Elbas covers the Nordic and the Baltic region as well as Germany and prices are set on a first come first served principle Nord Pool Spot n d The balancing market or day after marked is the result of deviations be tween planned produced consumed power and the actual production consumption of power at the delivery time causing imbalances in the electricity system In order to keep the system balanced at all times the system responsible operator buy up and down regulating power The following day accounts are settled between the system responsible and suppliers producers who caused the im balances Prices are determined according to a specific pricing model for the balancing power administrated by the system responsible The forward market is the mark
35. day ahead electricity price and the daily wind power production such that low production is associated with high price The most appropriate models for the dependency be tween filtered wind power production and filtered prices is the Normal copula alternatively the Frank copula e Extremely low wind power in feed does not cause extremely high day ahead electricity prices due to asymptotic independence That is ex tremely low wind power production occur independently from extremely high prices thus low wind power production can not explain the occur rence of extreme prices While there is a significant relation between excess wind power production and excess electricity prices no significant relation between prices and wind power production under extreme conditions in neither DK1 nor DK2 was found The results imply that the Danish power market so far has seen the positive price effects of the wind power as excess wind power production leads to lower prices At the same time the lack of asymptotic dependence show that low wind power production does not lead to extreme prices as the extremely high prices appear uncorrelated with extremely low wind power production While previous studies show that wind power increases electricity price volatility this study suggests that it does not cause price spikes as the upper tail behavior of the electricity prices remains independent from the wind power production In conclusion the Danish power m
36. dered By investigating the QQ plots of GEV distributions fitted to block maxima of different block lengths a reasonable block length can be determined The block maxima of WP residuals seem to follow GEV distribution for any choice of block length The two maximum values of the price residual block maxima are higher than expected for all block lengths indicating that the fitted GEV distributions do not capture the tail behavior However the return level plots show that for the block length of 40 days the two maximum observation lie within the 95 confidence interval of the return level This is not the case for shorter block lengths and with this reason the block length is set to 40 days The block length of 40 days result in 27 block maximum observations The density functions of the fitted GEV are plotted and compared with empirical density of the 40 day block maxima in Figure 21 Estimated copula parameters found using FML CML and IFM methods are presented in Table 14 together with goodness of fit test results Estimated GEV parameters of the margins can be found in Appendix G 54 Log Goodness of fit p value DEZ Capuli Pareles likelihood S test M test Gumbel 1 13 0 148 38 415 0 03946 0 5939 Tawn 0 4 0 378 38 885 0 03047 0 484 Husler Reiss 0 772 0 247 38 525 0 05045 0 6878 Galambos 0 445 0 199 38 138 0 05544 0 6479 Gumbel 1 232 0 193 0 9484 0 03846 0 6069 Tawn 0
37. e first and the latter give a full paramet ric description of both copula and marginal functions while the CML method uses empirical marginals when estimating the parameters of the copula Each method has advantages and disadvantages and consistency across parameter values resulting from the different methods indicates reasonable assumption of the marginal distribution functions With this reason all three methods are applied The methods are used when investigating both the full sample dependence structure and the extreme value dependence structure 19 Full Maximum Likelihood The Full Maximum Likelihood FML method estimates parameters of the copula as well as the marginal distribution func tions simultaneously by maximizing the full model s likelihood The copula density for a copula with parameters is in the bivariate case given as below Gudendorf and Segers 2009 82 O ux uy ux uy e 0 1 14 ux uy Co Ux uy The parametric marginal distributions Fx and Fy with unknown param eters y and o give the uniform margins The parameters of the copula and the marginal distributions is found as the arguments that maximize the log likelihood stated below 1 0 P 0 5 loge Fx Xi Fy Yi i l The FML estimation may require high computational power thus other less computer intensive methods are suggested Canonical Maximum Likelihood The Canonical Maximum Likelihood CML method consist of a
38. e residual o EE E ESI E A AE AE E EEEN EE nN eo o 6 L g a r I oot Nn gt 4 7 o gt EEEE ESEE TELE EEEE A SEEE ESEE E EER 7 T T T T T T T 0 5 10 15 20 29 30 Lag days d Sample ACF of the squared price residuals in DK1 resulting from fitting an ARMA 3 3 GARCH 1 1 model to the price series The dashed lines indicate the threshold for 95 sig nificance level No significant autocorrelation can be identified thus no heteroscedasticity Price filtering ee DK2 Price Filtering DK2 First difference daily electricity price Price diff DKK per MWh 200 0 fi fi 400 1 T 2012 2013 2014 2015 t year a First difference of daily electricity prices in DK2 DK2 ACF Price residual 0 02 1 ACF 0 02 0 06 Lag days c Sample ACF of the price residual in DK2 resulting from fitting an ARMA 4 3 GARCH 1 1 model to the price series As no ACF values surpass the dashed line which in dicates the threshold of significance at a 95 confidence level no significant autocorrelation can be identified Figure 27 DK2 66 DK2 Price residual 10 Price residual 0 10 2012 2013 2014 2015 t year b Standardized price residual for DK2 result ing from fitting an ARMA 4 3 GARCH 1 1 model with weekday indicator variables to first difference of prices DK2 ACF Squared price residual
39. ed capacity on the transmission line over the Great Belt combined with limited possibility for importing from Germany Negative prices occur in DK1 as well as in DK2 DK1 Daily Electricity Price DK2 Daily Electricity Price a a 2 8 T F A 8 4 F cy Oo a 2 a 87 34 ay pe a 8 S a o a 24 2 J S o4 T 2012 2013 2014 2015 2012 2013 2014 2015 t year t year Figure 2 Daily electricity prices in DK1 West Denmark and DK2 East Denmark Limited interregional transmission capacity resulted in price spike in DK1 on June 7 2013 In both DK1 and DK2 negative power prices have been observed 25 In this section the results of the ARMA GARCH modeling of the prices are presented The choice of the model will be further justified and the goodness of fit of the models will be discussed While there is no clear sign of yearly seasonality during the 35 5 months of observations the prices do show a weekly periodicity which supports the idea of including weekly periodicity in the model Moreover the price series do not look stationary suggesting that modeling the first difference of the power prices is appropriate 3 2 1 DK1 Price model in West Denmark The KPSS test for the first difference of price series indicates that the assump tion of stationarity cannot be rejected at a 95 confidence level Figure 3 shows the sample ACF of the first difference of daily prices in DK1
40. et for financial contracts used by producers and suppliers for risk management purposes The contracts are traded at the NASDAQ OMX commodities exchange and includes futures forwards and options using the system price as reference Moreover Contracts for Difference based on the difference between area prices and system price are traded In this thesis only the prices in the day ahead market will be considered and in the following price will refer to day ahead price Elspot Moreover the prices are given as the daily price which is calculated by Nord Pool Spot as the daily average 2 Method This chapter introduces the research structure as well as the employed meth ods The modeling consists of four parts First the electricity prices are filtered in order to remove seasonality weekly variations and other general trends in the prices The methods used for this purpose are presented in Section 2 1 Also a filter for the wind power production is applied as presented in Section 2 2 After modeling prices and wind power production the dependence struc ture is investigated with the help of copulas First the general dependence is examined using the copulas presented in 2 3 1 and finally the extreme value theory will help investigating the dependence structure of extreme events The extreme value approach is presented in 2 3 2 The structure of the research is inspired by the article Aloui et al 2014 which separates the investigation of
41. f no autocorrelation in the residuals on a 95 confi dence level with all p values greater than 0 05 Likewise the hypothesis of no ARCH effects in the residuals can not be rejected on a 95 confidence level by the Ljung Box test applied to the squared residuals and the ACF of the squared residuals are insignificant for all non zero lags thus no remaining het eroscedastic effects can be identified in the residuals ACFs and residual plots can be found in Appendix B Figure 5 shows the QQ plots for the price residuals in DK1 as well as DK2 The theoretical quantiles of the student t distribution are plotted against the empirical quantiles of the residuals The plots confirm that the assumption of student t distributed residuals is appropriate for both areas and only the tails of the residual distributions seem to deviate a little from the tails of the student t distribution In conclusion when using the estimated model as filter the remaining price residuals can be assumed to be independently student t distributed DK1 QQ plot for residuals student t DK2 QQ plot for residuals student t 10 o o 20 o Sample quantiles 10 Sample quantiles 0 i 10 10 o Theoretical quantiles Theoretical quantiles Figure 5 QQ plot of the price residuals of DK1 and DK2 repectively The assumption of student t distributed residuals is appropriate as the empirical quantiles plotted against the theoretical qu
42. for the prices of DK1 is ARMA 3 3 GARCH 1 1 the optimal model for DK2 is ARMA 4 3 GARCH 1 1 The models successfully remove autocorrelation and het eroscedasticity from the price series 2 The daily wind power is filtered using an ARMA GARCH model The optimal model of for the wind power in DK1 is ARMA 1 2 GARCH 1 1 and for the series in DK2 ARMA 3 3 GARCH 2 3 under the assump tion of normally distributed residuals The models remove all autocor relation and heteroscedasticity however residuals are slightly skewed 3 Copula fitting to the observations of price residuals and WP residuals in DK1 as well as in DK2 indicate a significant dependence between the price and WP residuals The Normal and Frank copulas have the highest likelihood and highest goodness of fit p value while the t copula could also not be rejected Linear correlation coefficients between price residuals and additive inverse wind power residuals are significant and positive 4 Extreme value modeling using block maxima with block lengths of 20 and 40 for DK1 and DK2 respectively indicates asymptotic independence None of the fitted copulas could be rejected using the Mpn test however all dependence parameters are close to those of independence x plots and x plots support upper tail independence 61 4 2 Conclusion The results of the study can lead to the following conclusions for both DK1 and DK2 e There exist a clear relation between the
43. g E Table 2 List of extreme value copulas used for investigating the extreme dependence denotes the cumulative standard normal distribution function that the copula C of Mn is given by C u v Oplu v3 If there exist a Cr such that the copula Cp gt C for n oo then C is called an extreme value copula Moreover it can be shown that the bivariate copula C is an extreme value copula if and only if Cut uaa eee Cu a e 0 1 0 1 11 where A 0 1 gt 5 1 is convex and satisfies max t 1 t lt A t lt 1 for all te 0 1 Gudendorf and Segers 2009 The function A t is called Pickands dependence function and is useful for understanding the dependence structure The upper bound A t 1 gives the independence C u v u v while the lower bound A t max t 1 t corresponds to perfect dependence resulting in the copula C u v min u v Pickands dependence function can also be used to calculate the upper tail dependence by x limu 41 P U gt ulV gt u 2 1 A 1 2 12 In this thesis four different types of EV copulas will be used for modeling the extreme value dependence and the four kinds is presented in Table 2 18 Modeling extremes with Block Maxima The GEV is the limiting dis tribution of the maximum that means the distribution of the maximum of an infinite number of observations Even if an infinite number of observations were available
44. gt gt 5 o S amp E o g P D D on ow Ei 4 2 5 20 100 500 2 5 20 100 500 Return Period 10 days Return Period 10 days Price residual BM WP residual BM 20 20 5 g S s E 2 E 2 E E p an ite 2 5 20 100 500 2 5 20 100 500 Return Period 20 days Return Period 20 days Price residual BM WP residual BM 30 30 9 Q a N oa E is pr 2 E g e Ew 2 ire 2 5 20 100 500 2 5 20 100 500 Return Period 30 days Return Period 30 days Price residual BM WP residual BM 40 40 3 8 a gt gt ig S E o E p T D Erg x 2 5 20 100 500 2 5 20 100 500 Return Period 40 days Return Period 40 days Figure 20 Return level plots for price residuals and wind power residuals for different block lengths in DK2 Dashed lines indicates the 95 confidence interval Only for the block length of 40 days the maximum price residual lies within the confidence interval 53 DK2 Density of price res BM DK2 Density of WP res BM 1 5 ae Empiricg H Empirica i Modeled A Modeled 1 0 Density 0 2 04 Density 05 fi 0 0 L 0 0 N 27 Bandwidth 0 2284 N 27 Bandwidth 0 1615 Figure 21 Density plots comparing the density of the fitted GEV distribution to the empirical density of the 40 day block maxima of price residual and and WP residual in DK2 Since block lengths larger than 40 days will result in very few observations such block lengths are not consi
45. have no autocorrelation up to lag K the distribution of the test statistics Q can be approximated with a gamma distribution for which the shape and scale parameters are given as 3 K k 1 a 42K 3K 1 6K p q 12 as E tO 3 K K 1 For values of Q lt T o 95 a 3 the residuals show no significant autocorrelation and the model can be accepted Homoscedasticity The residuals can be checked for remaining heteroscedas tic effects by investigating the squared standardized residuals In case the residuals exhibit heteroscedasticity the squared residuals will show autocorre lation The sample ACF of the squared residuals is useful for identifying au tocorrelation at individual lags and with a Ljung Box test applied to squared standardized residuals the hypothesis of homoscedasticity can be tested The test statistics is given as Bi Q n n 2 gt gt 2 where 7 is the estimated k lag autocorrelation of the squared residuals Q is y distributed with K degrees of freedom under the null hypothesis implying that the null hypothesis of homoscedasticity is rejected on a 5 confidence level if Qs gt X 95 K 2 2 Wind power model The wind power production depends on the weather conditions During pe riods with little wind the generation is low compared to periods with high wind speeds and since the weather conditions one day is highly related to the weather the previous days so is the wind power product
46. he 50 Log Goodness of fit p value DEE orn PORTE E Test Gumbel 1 13 0 132 84 17 0 5779 0 2782 Tawn 0 21 0 36 84 43 0 3561 0 1434 m Husler Reiss 0 50 0 272 84 10 0 6538 0 3182 Galambos 0 27 0 17 84 03 0 5989 0 3122 Gumbel 1 1395 0 1153 0 9174 0 5629 0 2542 Tawn 0 2678 0 2482 0 5425 0 3731 0 1673 Husler Reiss 1 1392 0 1152 0 9174 0 6908 0 3082 Galambos 0 3958 0 1387 1 109 0 6139 0 3112 Gumbel 1 0668 0 1018 0 2439 0 55 0 2612 Tawn 0 1579 0 2786 0 1508 0 3851 0 1703 Husler Reiss 0 5108 0 2200 0 1386 0 6858 0 3292 Galambos 0 2551 0 1622 0 1729 0 6159 0 2822 Table 13 Goodness of fit test results for extreme value copulas fitted to BM of DK1 using FML CML and IFM None of the models can be rejected on a 95 confidence level as all p values are greater than 0 05 FML estimation Likewise the CML gives parameter estimate close to the parameter values of the independence copula While none of the models can be rejected on a 95 confidence level the Galambos copula has the highest log likelihood Inference functions for marginals The estimation results using the IFM method are similar to those of the other two methods Parameters are close to the values leading to independence As for the other estimation methods the Tawn copula gives the worst goodness of fit however n
47. he maximum of the additive inverse min X X Xn max X1 Xo Xn In the multivariate case the maximum can be defined as the vector of com ponentwise maxima Given the d dimensional series X Xa Xi2 Xia for i 1 2 n the maximum is defined as Mp Ma Mio Mad where Mag max X1 X9 sgn In this thesis only bivariate data will be used For bivariate data the expression of the maximum can be reduced the following expression given the series X1 Y1 X2 Yo Xn Yn Mhp Mix Mny 10 The problem of finding the bivariate joint distribution of the two maxima is equivalent to finding the marginal distributions and the copula By recognizing that the componentwise maxima have the univariate GEV as limiting marginal distributions the problem is reduced to finding the copula connecting the margins Let F be the joint distribution of X and Y with marginal distributions Fx and Fy and the copula Cr The maximum of n pairs X Y is defined by equation 10 with joint distribution F and marginals FY and FY It follows 17 Extreme value copulas Name C u v Parameter Independence 1 0 Gumbel exp logu logu 1 lt 0 lt 0 1 Tawn uvexp 0 er lose 0 lt 68 lt 1 0 0 Galambos uvexp logu logv 0 lt lt 0 0 1 1 logu Husler Reiss logue Tar a 4 0 lt 8 lt 0 0 logu 4 6lo
48. ikewise conclude that renewable energy reduces the spot electricity prices Both Ketterer and Paraschiv et al assume linear relation between price level and volatility and wind power in feed If price volatility increases with the share of wind power in the power system larger price spikes would be expected in markets with large wind power generation Some researches relate the market volatility price level and spike probability to the share of renewable energy in the power sys tem by comparing power markets of different regions however inference made by comparing markets may be disturbed by fundamentally different market conditions which cannot be accounted for in the models The approach pre sented in this thesis aims at relating the price spikes directly to the daily wind power production inside the given market without assuming linear dependence 4 Figure 1 Danish transmission net Energinet dk n d 1 2 The Nordic electricity market In this section the basic concepts of the electricity market will be presented Denmark is a part of the Nordic Baltic electricity market which besides Denmark includes Norway Sweden Finland Estonia Latvia and Lithuania The transmission systems of the Nordic countries are interconnected and the Danish power grid is connected with the Swedish grid to the east and the Norwegian grid to the north but also the German grid to the south Ea En ergianalyse A S 2005 The power system of Denmark
49. ikls Latvia and has a market share of 84 of the total power consumption in the Nordic Baltic market as of ultimo 2013 Nord Pool Spot 2013 1 2 1 Wholesale power market This thesis will focus on the wholesale market and in the following the concepts necessary for understanding the wholesale market will be introduced Whole sale market is the place where power producers and suppliers trade power and it is said to consist of four phases or submarkets e Day ahead market e Intraday market e Balancing market e Forward market The day ahead market includes the power traded between suppliers and pro ducers for covering the power production and consumption the following day The day ahead power prices are usually referred to as spot prices and the power to be consumed in the Nordic Baltic region including Denmark is traded under the name Elspot on the Nord Pool Spot exchange The Nordic region consist of several bidding areas in Denmark DK1 and DK2 and for each area and each hour a price is set on the market The Elspot area price is determined by the 6 equilibrium between the aggregated supply and demand curves for each of the bidding areas and hour with respect to the transmission capacity between each area As a reference price for the Nordic region the system price is also calcu lated from the sale and purchase bids disregarding the available transmission capacity between the bidding areas The system price is used for trading and
50. ion Like the elec tricity prices the wind power production series also exhibits heteroscedasticity Due to autocorrelation and heteroscedasticity the wind power production is also modeled as an ARMA GARCH process with the model specifications as below p q AWP amp 5 bo AW Py_ y jeti 4 i 1 j 1 13 P Q it 1 w gt Biot acy p gt O51 gt i 1 j l Ct Otjt 1Et Here the AWP is given as the first difference of the wind power production AWP WP WP The procedure for selection of model order parameter estimation and model diagnostics are equivalent to that of the price model presented in 2 1 1 2 3 Dependence modeling Once the electricity standardized price residuals and the standardized wind power production residuals are found the dependence between them can be modeled First the dependence for the entire sample is investigated to reach conclusions on the general relation between price and wind power production For this purpose the notion of copulas will be introduced in Section 2 3 1 Secondly the extreme value dependence will be examined using the component wise block maxima The bivariate extreme value theory introduced in Section 2 3 2 will be used to investigate the dependence between the maximum price residual and minimum wind power residual 2 3 1 Copulas Copulas are multivariate joint distributions functions whose one dimensional marginal distributions are uniform The d dimens
51. ional copula is given as C u1 U2 5 Ud 1 Ua POLS u1 U2 lt u2 Ug lt Ua 1 Ua lt Ua where U are uniform 0 1 distributed In equation 6 below the two dimensional copula is presented A more formal definition can be found in Nelsen 2004 C u v PU lt u V lt v 6 14 The most important characteristics of the two dimensional copula are C u 1 u C 1 v v C u 0 0 C 0 v W u v lt C u v lt M u v where W u v maz u v 1 0 and M u v min u v The inequality above is called Fr chet Hoeffding bounds inequality In case of independence between the variables the formulation can be re duced to the independence copula C u v uv 7 By applying Sklar s theorem the theory of copulas becomes useful for mod eling dependence structures between random variables For the 2 dimensional continuous case Sklar s theorem can be expressed as Let F be a joint distribution function with margins Fx and Fy Then there exists a copula C such that for all x y in R F x y C Fx Fy y 8 If Fx and Fy are continuous then C is unique If C is a copula and Fy and Fy are distribution functions then the function F defined by 8 is a joint distribution function with margins Fy and Fy Nelsen 2004 According to Sklar s theorem the copula contains all information about the de pendence structure and the task of modeling the joint distribution of random variables c
52. is divided into two areas West Denmark DK1 and East Denmark DK2 separated by The Great Belt The two areas were physically connected by a transmission line put into operation in August 2010 Figure 1 provides an overview over the Danish transmission net The structure of the electricity market can be summarized with the follow ing Power producers sell the electricity on the whole sale market to suppliers who then sell the electricity on the retail market to the final consumer The delivery of the electricity is ensured through the transmission and distribution network The Nordic electricity markets has been greatly liberalized during the last two decades opening both electricity trading and production to competition Liberalization introduced competition in the whole sale as well as the retail market While the retail market liberalization for instance gave Danish con sumers the right to freely chose their power supplier the whole sale market liberalization has given way for the opening of the common Nordic power ex change Nord Pool Spot in which producers and suppliers trade power Nor dReg 2014 Nord Pool Spot former Nord Pool was established as a part of the Nordic market integration and is based in Oslo Norway It is owned by the national transmission system operators Statnett SF Norway Svenska Kraftnat Swe den Fingrid Oyj Finland Energinet dk Denmark Elering Estonia Lit grid Lithuania and Augstprieguma t
53. ist significant autocorrelation for lag 7 14 21 and 28 indicating the need of adding weekday indicator variables in the model Right ACF of squared first difference power prices in DK2 The dashed line indicates the threshold value for significance at a 95 confidence level 3 2 2 DK2 Price model for East Denmark The analysis of the price series in DK2 is equivalent to that of DK1 The KPSS test cannot reject null hypothesis of stationarity on a 95 confidence level ACF of the first difference of daily prices is presented in Figure 4 in which evidence for weekly periodicity is found in terms of the clear 7 day pattern in the ACF including significant autocorrelation at lag 7 14 21 and so on Negative autocorrelation for lag 1 is the result of mean reversion in the price series however unlike the price series of DK1 significant autocorrelation can be found for other lags than the first Significant values of the ACF of the squared first difference of prices indicate heteroscedasticity and justifies the need of using GARCH specifications for the conditional variance Model selection The model resulting in the lowest AIC value was the ARMA 4 3 GARCH 1 1 model with weekday indicator variables and student t distributed residuals The estimated parameters are presented in Table 5 29 Parameter Estimated value i 2 131264 OP 1 659532 AR 03 0 420233 Q4 0 065418 0 2 539241
54. la gives the best goodness of fit regardless of estimation method 42 Full maximum likelihood The results of the FML estimation show that all copulas dependence parameters are far from those of independence given in Table 1 The dependence parameter of the normal copula equal to the linear correlation parameter is estimated as 0 64 and is significantly different from 0 This indicates a non zero positive correlation between the variables Table 11 shows the results of the goodness of fit test using Sn as the test statistic Of the five fitted copulas the Frank normal and t copula can not be rejected as appropriate model at a 95 confidence level Notice that the estimated degrees of freedoms of the t copula is very high thus the fitted t copula is very similar to the normal copula When taking the estimator variance into consideration the estimates of the parameters of the marginal functions are consistent throughout all copula models Canonical maximum likelihood The dependence parameters agree with those estimated with FML Again all parameters are far from the values leading to independence and the Normal Frank and t copula cannot be rejected on a 95 confidence level by the goodness of fit test Inference functions for marginals Results of the IFM estimation is also presented in Table 11 As in the previous estimations all parameters are signif icantly different from the parameters of independence in the respective models M
55. las The aim of the thesis is to investigate the extreme value dependence between the electricity prices and the wind power production Extreme value theory introduces the framework for modeling the maximum or minimum of variables using the limiting extreme value distribution The general ideas of the extreme value theory are presented below First the univariate case is considered as described in Coles 2001 For a series of IID stochastic variables X1 X2 Xn with the common distribution Fy the maximum M is defined as My max X X Xn The distribution of M can then be expressed in terms of Fy PUGS P X1 lt z X2 lt z Xn lt 2 P X lt 2 FX 2 i 1 16 Extreme value theory deals with the limiting distribution for n gt oo It can be shown that given the existence of normalization parameters an and bn Mn bn an the limiting distribution of is a so called Generalized Extreme Value distribution GEV expressed in equation 9 ao p 7 lt 4 a Ga EN 9 The distribution is defined for the set z 1 y z u o gt 0 while the pa rameters satisfy oo lt u lt o gt 0 and lt y lt oo The limiting distribution takes the form of 9 regardless of the distribution Fy By using the additive inverse of the data series the GEV as in equation 9 can be used for modeling of the minima This is the result of the minimum being expressed in terms of t
56. ls on the kth day of the week The model presented in equation 2 will in the following be referred to as the mean equation GARCH While the mean equation aims at filtering away autocorrelation i price series the volatility clustering of the prices also needs to be taken into account in the model The price volatility depends on the general market conditions and may vary over time which means that some periods have low volatility while others may have high The implication of conditional volatility is that the errors e may not be identically distributed Volatility clustering heteroscedasticity can be modeled using a general ized autoregressive conditional heteroscedasticity model GARCH With the GARCH model variance is modeled as conditional on previous observations The GARCH P Q model is presented in equation 3 below P Q Oit w gt D Biotec D OC 3 i l j ae Ct Otjt 1Et 10 By taking the changing volatility into consideration the resulting stan dardized residuals can be considered IID and be used for modeling the dependence structure with wind power production Estimation method The model parameters to be estimated are for i 1 2 p 0 for j 1 2 q Cr for k 1 2 7 i for i 1 2 P and a for i 1 2 Q The parameters of the ARMA GARCH model can be found as the ar guments to the maximum log likelihood ML The maximum likelihood of ARMA GARCH model
57. ly not However in this thesis return measures are not used due to the occurrences of negative prices The logarithm is not defined for negative values thus the log return series would have missing val ues whenever negative prices are observed The return on the other hand may be numerically calculated however it does not reflect the price structure in an appropriate manner Under normal conditions of positive prices positive returns are connected with price increase and negative returns are connected with a decrease in price However negative prices challenge this simple rela tionship If for instance the negative price P _ is followed by a positive price P the resulting return will be negative even though the price increased The problem of negative prices is the main reason for not transforming prices into returns or log returns in this thesis Moreover the concept of return is well understood for classic financial assets such as a stock which can be bought for P at time t 1 and be sold for P at time t resulting in a return For electricity prices return does not have a similar interpretation as the electric ity cannot be stored The price P is simply the price of the electricity to be consumed at time t and not the price of a given asset at time t Stationarity The power price series are not stationary In order to obtain stationarity the first difference of the power prices are used instead of the price itself AP P
58. nce bandga 5 N4 E 4 oO oO 3 o J 3 o Leg 2 ot a a E oO of oO u _ i N a a r D oo pa You A O e s Eo T T T T T T T T T T T 4 2 0 2 4 4 2 0 2 4 Theoretical quantiles Theoretical quantiles Figure 13 QQ plot before and after replacing the minimum and maximum values in DK1 DK2 QQ plot for residuals student t DK2 QQ plot for residuals adjusted tail student t g zeeseese 1 1 line Sees Ee ae fis asl regression line E g regression line oe 95 confidence bg 95 gpr Sample quantiles 0 fi Sample quantiles 0 fi 5 l 10 Theoretical quantiles Theoretical quantiles Figure 14 QQ plot before and after replacing the minimum and maximum values in DK2 Figure 13 and Figure 14 show the QQ plots before and after tail corrections for DK1 and DK2 respectively In the plots the residuals empirical quantiles are compared to the theoretical quantiles of the student t distribution While the maximum and minimum residual observations before correction lie far from the theoretical quantile values the adjustment bring the outliers close to the 1 1 line Al It is worth noticing that the altering of the value makes no difference for the empirical margins used in the CML as the observations keeps their ranks in the series 3 4 1 DK1 Dependence structure in West Denmark The estimated copula parameters resulting from fitting the five copulas to residuals
59. ns from empirical non parametric uniform transforms in DK1 and DK2 respec tively DK2 Non parametric uniform transform v WP residual u Price residual 39 Tail correction for price distributions in DK1 and DK2 The assump tion of the distribution of the price residuals was validated with the QQ plots in Figure 5 for both DK1 and DK2 However the plots show that the assumption only holds for the large midsection of the observations and not for the tails as the t distribution has flatter tails than the observed distributions of the resid uals Thus assuming t distribution for the price residuals has the consequence that the probability of observing the most extreme residual values is numeri cally zero The probability of zero results in infinite negative log likelihood of the assumed model which leads to error during the log likelihood optimiza tion This error indicates that the specified parametric marginal distribution is insufficient for capturing the tail behavior however since no better fitting marginal distribution is found the numerical optimization problem is solved by correcting the most extreme values in the price residual series For instance in DK1 the residual observation related to the extreme electricity price on June 7 2013 is replaced The replacement is based on extreme value theory and done according to the expected return levels which allow the replacement values to be extreme but still occ
60. ock length gives only a few observations thus may result in large variance The block length is chosen for DK1 and DK2 as the shortest length for which both block maxima price residuals and block maxima WP residuals seem to fit the generalized extreme value distribution well For DK1 the block length is set to 20 days and for DK2 40 days and the reason behind choosing these particular block lengths can be found under the analysis of the respective areas DK1 Componentwise block maxima DK2 Componentwise block maxima o o 2o o o T w o wo o g za o 2 o g ae g 2 o o 5 o S S D o ro Bem 47 8 e o o p o D 3 o o Pes D 8 o D o Q o v o 7 o o Ea o Q o o Q o wo of 24 9 oaee A o o o o T T T T T T T T T T 0 1 2 3 4 5 2 4 6 8 Price residual Price residual Figure 15 Block maxima price residuals vs block maxima wind power residuals of DK1 and DK2 respectively The sample of DK1 has more observations than DK2 due to the shorter block length Observe that x axis in the DK1 plot has been cropped such that maximum price residual is outside the plot 47 DK1 BM non parametric uniform transform DK2 BM non parametric uniform transform oOo e3 o o o o o e o o o o o J 9 o o 9 o o o o co o o co o 5 3 o D Jg oi my OL o o o o a on Oo o 2 o 2 2 o Q o a o o x p a o a Q o
61. one of the copula models can be rejected 3 5 2 DK2 Extreme dependence in East Denmark For the data series of DK2 the block length is set to 40 days and this choice is based on the following analysis In Figure 19 the QQ plot for block maxima price residuals as well as block maxima WP residuals are shown for block length of 10 20 30 and 40 days Figure 20 show the return level plots for same blocklengths 51 Price residual BM WP residual BM 10 10 S 8 o ea S E 6 D a a t Be 5 5 o 2 amp 2 S 140 15 20 25 Model Quantiles Model Quantiles Price residual BM WP residual BM 20 20 E E 2 we kE Boa x 6 B a 2 g s a N a E E w w 1 z 3 4 3S 1 5 2 0 25 Model Quantiles Model Quantiles Price residual BM WP residual BM 30 30 E 2 p goo Bo Gg O a a E 2 a a ai E w w 2 3 lt 5 12 48 20 24 Model Quantiles Model Quantiles Price residual BM WP residual BM 40 40 Empirical Quantiles 2 4 6 8 Empirical Quantiles 16 20 26 2 3 4 5 14 18 22 26 Model Quantiles Model Quantiles Figure 19 QQ plots for block maxima price residuals and wind power residuals for block length 10 20 30 and 40 days in DK2 While the two maximum price residuals seem to lie outside the GEV distribution for all of the plotted block lengths the block maxima wind power residuals follow the GEV already for a block length of 10 52 Price residual BM WP residual BM 10 10 oo
62. or parametric models of the margins unlike the CML 2 3 4 Model selection and verification In order to accept or reject the parametric copula model goodness of fit tests are used As suggested by Genest et al 2006 the test statistic S can be used to measure how close the estimated parametric copula is to the empirical copula The test statistics is defined as Dan f GRC eC A E 17 21 where Co is the parametric copula and C is the empirical copula C is given in terms of the uniform pseudo observations 6 as in equation 16 Cruzu I i lt u i lt v 1 n Sle For the extreme value copulas a goodness of fit test based on Pickands dependence function can be applied The test statistic M is defined as in Genest et al 2011 i I A t Ag t Pat 18 where A t is the empirical Pickands function and Ag t is the Pickands function resulting from the parametric model of the copula The empirical Pickands function is estimated with the Cap raa Fougeres Genest method as presented in Cap raa et al 1997 The p value associated with the test statistics can be calculated with para metric bootstrap however as this method is very computer intensive the multiplier approach presented by Kojadinovic and Yan 2010 is used for the S test 2 3 5 Identifying independence An important question in relation to the investigation of the dependence struc ture between the price and wind power is ho
63. oreover the parameters agree with the values estimated under FML Again the result of the goodness of fit test shows that Normal Frank and t copula fit well enough not to be rejected on a 95 confidence level 3 4 2 DK2 Dependence structure in East Denmark The analysis of the dependence between price and wind power in DK2 is equiv alent to the one of DK1 Following the same procedure as in the previous section the five copulas in Table 1 are fitted to the full sample data of price residuals and WP resid uals Main results are presented in Table 12 and the estimated parameters of marginal functions using FML and IFM are presented in Appendix F 43 Log Goodness of fit DK2 Copula Parameter likelihood om TAR Normal 0 5803 0 02 2723 359 0 0249 0 1364 Clayton 0 9498 0 07 2761 503 0 3432 0 0004995 Gumbel 1 597 0 045 2757 028 0 1856 0 0004995 a Frank 4 206 0 222 2731 662 0 0271 0 06543 0 5813 0 021 t 48 39 35 85 2723 762 0 0256 0 1404 Normal 0 5852 0 0174 222 3 0 0249 0 1424 Clayton 0 9064 0 0566 179 3 0 3432 0 00050 Gumbel 1 554 0 0370 185 8 0 1856 0 00050 O Frank 4 180 0 2083 212 9 0 02171 0 0764 0 5844 0 01756 ie 111 87 209 33 222 2 0 0257 0 1284 Normal 0 5793 0 0175 221 6 0 0249 0 1324 Clayton 0 8984 0 0559 180 3 0 3432 0 00050 Gumbel 1 543 0 0366 181 6 0 1856 0 00050 Frank 4
64. pendence y plot and y plot constructed on the basis of the price residual block max ima and wind power residual block maxima are presented in Figure 23 The functions have large variances due to the few observations The y plot does not reject the assumption of the maxima being independent as the x u function does not significantly depart from 0 Moreover the x u is very far from 1 as u 1 thus there is no indication of upper tail dependence Like the y plot and x plot for DK1 the plots for DK2 indicate a significant dependence between the price residuals and the wind power residuals The x plot and x plot for DK2 can be found in Figure 24 The x u function value is greater than zero for all u however the function is decreasing and reaching values close to 0 for u 1 which indicates asymptotic independence x u is smaller than 1 for u 1 thus the x plot supports the idea of asymptotic independence In Figure 25 the y plot and x plot for the price and wind power residual block maxima can be found Here the y u function value is not significantly different from 0 for any value of u thus independence between the maxima is indicated Long block length and few observations result in large estimate variances thus reaching conclusion on the basis of the block maxima y plot and y plot is not possible E g asymptotic independence cannot be rejected as 1 lies side the confidence interval of x u for u gt 1 57
65. r optimizing the model s log likelihood thus parameter estimates may be suboptimal The significance of the skewness cannot be rejected in neither DK1 nor DK2 however since the skewness is numerically small no further investigation on the skewness will be conducted In Figure 10 the residual empirical distributions are compared to skewed normal distributions In conclusion the WP residuals resulting from filtering the first difference WP series with the specified models can be assumed IID and in the further analysis the WP residuals are modeled with the skewed normal distributions 36 DK1 QQ plot for residuals normal DK2 QQ plot for residuals normal o o GOO Sample quantiles Sample quantiles 2 L 4 1 a 1 Theoretical quantiles Theoretical quantiles Figure 9 QQ plots for the WP residuals in DK1 and DK2 under the assump tions of normality DK1 QQ Plot for residuals skewed normal DK2 QQ plot for residuals skewed normal oO o a a 4 P q os 3 G ej 2 4 a ee a E E oO oO on on y 4 y o 4 o o i g o Yoo T T T T T T T T T T T T 3 2 1 0 1 2 3 2 1 0 1 2 Theoretical Quantiles Theorectical Quantiles Figure 10 QQ plots for the WP residuals in DK1 and DK2 under the assump tion of skewed normality As the skewed normal distribution results in obser vations forming straight lines it is concluded that the residuals are slightly
66. reme value copula to the observations in DK2 are also presented in Table 14 As the FML estimation the CML gives parameter estimate close to the parameter values of the independence copula The Gumbel Tawn and Husler Reiss copula are rejected by the S test on a 95 confidence level while the M test does not reject any of the models 59 Inference functions for marginals The estimation results using the IFM method are similar to those of the other two methods Parameters are close to the values leading to independence and the Husler Reiss copula gives the highest log likelihood and p values in the goodness of fit tests however none of the models can be rejected according to the M test The S test rejects all but the Husler Reiss copula on a 95 confidence level 56 3 5 3 y plots and y plots The x plot and x plot for DK1 is presented in Figure 22 With the x u function value significantly above zero the idea of positive dependence be tween price residual and additive inverse of wind power residual is supported According to the theory of the plots presented in Section 2 3 5 upper tail in dependence exists if x u 0 for u 1 The y plot does show decreasing function value however it is not possible to determine whether function limits to zero or instead to a positive value near zero In the y plot on the other hand it is clear that y u does not limit to 1 as u gt 1 This gives a strong indication of asymptotic inde
67. rice residuals and the additive inverse of WP residuals in DK1 and DK2 Three methods are employed full maximum likelihood FML canonical maximum likelihood CML and inference functions for margins IFM The FML and IFM methods use parametric marginal distribution func tions with which the likelihood of the observations can be evaluated For this reason the parametric marginal distribution functions need to be speci fied During the fitting of the ARMA GARCH models to the electricity prices of DK1 and DK2 in Section 3 2 the price residuals were assumed student t distributed The wind power residuals are assumed to be skewed normally distributed as this distribution fits the residuals better than the symmetric normal distribution according to the conclusion of Section 3 3 1 In Figure 12 the copula based on empirical non parametric margins in DK1 and DK2 are presented 38 DK1 Price residual vs WP residual o 890 1 i fe WP residual additive inverse 2 fi o E L Price residual Figure 11 Price residuals vs additive inverse of wind power residuals in DK1 DK2 Price residual vs WP residual WP residual additive inverse Price residual and DK2 respectively The plots suggest a positive correlation DK1 Non parametric uniform transform v WP residual u Price residual Figure 12 Price residual vs wind power residual using pseudo observatio
68. s can be computed using a state space formulation and a Kalman filter to recursively generate prediction errors and their variances and then use the prediction error decomposition of the likelihood function The detailed description of the method can be found in Cryer and Chan 2008 The log likelihood is calculated using a prespecified residual distribution The assumption of distribution is determined experimentally as the one fitting the residuals more accurately In this thesis the choice of residual distribution is between normal and student t distribution 2 1 1 Model selection Akaike information criteria The models for electricity prices and wind power production will be selected according to the Akaike s Information Cri terion AIC The criterion states that the preferred model is the one that minimizes AIC 21 2k where l is the maximum log likelihood and k is the number of parameters in the model While the likelihood increases with the number of parameters in the model the AIC helps to avoid selecting a model with too many parameters thus ensures selection of a parsimonious model Autocorrelation The goal of fitting models to the price as well as the wind power time series is to filter out IID residuals which includes residuals being free of autocorrelation The autocorrelation of the price series is investigated using the sample autocorrelation function ACF Significant autocorrelation indicates the need 11
69. s equal to the linear correlation coefficient is around 0 63 in DK1 while it is 0 58 in DK2 and both values are significant according to the likelihood ratio test How ever the difference between the correlation estimated for the two areas is not statistically significant In the next section the extreme dependence will be investigated Accepting the Normal and Frank copula as appropriate models for the full sample depen dence an idea about the extreme dependence can already be established using the definition of upper tail dependence given in equation 12 The extreme dependence of the Normal copula can be shown to be zero except in the case of perfect dependence Likewise the Frank copula has upper tail dependence of zero 46 3 5 Estimation of extreme value copula and extreme de pendence structure The results of the investigation of extreme dependence that is the dependence between the maximum values of the price residuals and additive inverse of WP residuals are presented in this section It should be noted that for this analysis the price residual series without tail corrections are used First an appropriate block length is identified in order to collect a compo nentwise block maxima sample from the full sample A short block length gives a large sample which in return gives small variance of the estimates however the results may be biased as the sample does not represent the maximum well On the other hand a large bl
70. sformation Scandinavian Journal of Statistics Volume 33 2 pages 337 366 Gudendorf G and Segers J 2009 Extreme Value Copulas Joe H and Xu J J 1996 The estimation method of inference functions for margins for multivariate models Technical Report no 166 Department of Statistics University of British Columbia 63 Ketterer Janina C 2014 The impact of wind power generation on the electricity price in Germany Energy Economics Volume 44 July pages 270 280 Kojadinovic Yan and Holmes 2011 Fast large sample goodness of fit tests for copulas Statistica Sinica Volume 21 2 pages 841 871 Lindstr m E and Regland F 2012 Modeling extreme dependence between European electricity markets Energy Economics Volume 34 4 pages 899 904 Nelsen R B 2004 An Introduction to Copulas 2nd ed Nord Pool Spot n d About us Available from http www nordpoolspot com About us visited 2015 04 20 Nord Pool Spot 2013 Central to European Power Integration Annual Report Nord Pool Spot n d Day ahead market elspot Available from http www nordpoolspot com TAS Day ahead market Elspot visited 2015 04 20 NordReg 2014 An overview of the Nordic Electricity Market Available from http www nordicenergyregulators org about nordreg an overview of the nordic electricity market visited 2015 04 20 Paraschiv Erni and Pietsch 2014 The impact of renewable energies on EEX day ahead electrici
71. ta is appropriate The ACF of the squared first difference WP has significant values for some lags indicating heteroscedasticity in the data series This supports the choice of GARCH model for the conditional variance 32 DK1 ACF First difference daily WP DK1 ACF Squared first diff daily WP aan wo o a aen e e o e 2 LL i UN eee g H t e E E E S a Seo oe E eee eee E O EMENENM bg 6 8 dhd h A i w o y o sa i apa ape ine Se a ee ia 97 57 Fa wo e 31 T T T T T T T p T T T T T T T 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag days Lag days Figure 7 Left ACF of the first difference of daily wind power produced in DK1 The dashed lines indicate significance threshold for a 95 confidence level Significant autocorrelation is seen for lag one and two Right ACF of the squared first difference of WP in DK1 Parameter Estimated value AR Q 0 21306587 0 0 63528845 Ma 03 0 31117248 wW 2 21408162 GARCH a 0 02306379 p 0 96517653 Table 7 Parameter estimates from the fitting ARMA 1 2 GARCH 1 1 to the daily wind power production in DK1 Model selection According to the AIC the model to be selected for the WP in DK1 is the ARMA 1 2 GARCH 1 1 with normal distributed residuals The estimated parameters are presented in Table 7 The sample ACF of the residuals shows no sign of further autocorrelation as no ACF values are
72. th IFM method for observations from DK1 71 H DK2 Extreme dependence Below is the estimated parameters of the marginal functions estimated using the full maximimum likelihood FML method and the inference functions for margins IFM method Both block maxima of price residuals and wind power residuals are assumed to be GEV distributed DR Marginal function Price Marginal function WP Location Scale Shape Location Scale Shape Gumbel 1 852 0 114 0 524 0 104 0 428 0 169 1 654 0 059 0 256 0 047 0 097 0 214 Tawn 1 852 0 115 0 524 0 106 0 428 0 165 1 654 0 059 0 256 0 047 0 097 0 211 Husler Reiss 1 830 0 114 0 552 0 119 0 359 0 163 1 653 0 060 0 244 0 043 0 123 0 214 Galambos 1 848 0 114 0 524 0 105 0 451 0 177 1 654 0 059 0 255 0 047 0 101 0 214 Table 21 Estimated parameters for marginal functions using FML The block maxima observations are from DK2 Parameter estimates standard variations are given in parentheses Marginal functions Location Scale Shape Price residual BM 1 852 0 523 0 427 WP residual BM 1 654 0 256 0 0968 DK2 Table 22 Marginal function parameter values estimated with IFM method for observations from DK2 72
73. two step estimation where the observations first are transformed into uniform margins using the empirical distributions The parameters of the copula are estimated in the second step using maximum log likelihood The empirical uniform margins are based on rank and given by 1 2 Bay ee 15 where J is the indicator function In other words for the observation x of the time series X at time t the corresponding uniform pseudo observation of the empirical margin is i where i is the rank order of the observation x and n is the total number observations 20 The parameters of the copula are then found as the arguments that maxi mize the pseudo log likelihood given as 1 8 logcol x Xi Yi i 1 where the copula density cg is defined in 14 Inference Functions for Margins The basic idea of the method Inference Functions for Margins IFM proposed by Joe and Xu 1996 is to separate the estimation of the marginal parameters from the parameters of the copula In the first step the parameters of the marginal functions are estimated by the arguments that maximize the marginal functions log likelihood gand 6 Wp Slog fcg X i 1 Wer a i 1 where fx and fy are the marginal density functions The parameters of the copula are then estimated by maximizing the log likelihood 1 8 3 loge Fxe X FraYd i 1 This approach is computationally less intensive that the FML but still allows f
74. ty prices Energy Policy Volume 73 October pages 196 210 Rasmussen J N 2013 Vestdanmark far fredag h je elpriser i fem timer Available from http energinet dk DA El Nyheder Sider Vestdanmark faar fredag hoeje elpriser i fem timer aspx visited 2015 04 20 Aloui R et al 2014 Dependence and extreme dependence of crude oil and natural gas prices with applications to risk management Energy Economics Volume 42 March pages 332 342 64 Appendix A DK1 Price filtering DK1 First difference daily electricity price Price diff DKK per MWh 0 200 400 T T 2012 2013 2014 2015 t year a First difference of daily electricity prices in DK1 DK1 ACF Price residual E Le eee ee O O 5 l ull re Eo ee ee l F E ain ie nime et wd i ani tin F T T T T T T T 0 5 10 15 20 25 30 Lag days c Sample ACF of the price residuals in DK1 resulting from fitting ARMA 3 3 GARCH 1 1 model with weekday indicator variables to the price series The dashed lines indicates the threshold for 95 significance level No significant autocorrelation is seen Figure 26 DK1 65 DK1 Price residual 20 Price residual 10 10 T 2012 2013 2014 2015 t year b Standardized price residual for DK1 after fitting the ARMA 3 3 GARCH 1 1 model with weekday indicators to first difference of prices DK1 ACF Squared pric
75. ues estimated with IFM method for the observations in DK1 Price residuals are assumed t distributed while wind power residuals are skewed normal 69 F DK2 Full sample dependence Below is the estimated parameters of the marginal functions estimated using the full maximimum likelihood FML method and the inference functions for margins IFM method Price residuals are assumed t distributed while wind power residuals follow the skewed normal distribution DK9 Marginal function Price Marginal function WP Mean SD DF Mean SD Skew Normal 0 0054 0 026 0 976 0 039 4 5837 0 610 0 0013 0 029 0 997 0 022 0 711 0 029 0 028 Clayton 0 0407 0 025 1 048 0 055 3 7557 0 445 0 0171 0 029 1 012 0 021 0 655 Frank 0 0022 0 025 0 968 0 034 5 2420 0 807 0 0061 0 030 1 010 0 023 0 689 0 030 0 025 0 026 0 977 0 039 0 029 0 039 0 029 0 055 0 028 Gumbel 0 0005 0 026 1 046 0 053 3 8830 0 478 0 0017 0 030 1 009 0 022 0 779 0 032 0 034 0 030 0 039 0 029 t 0 0064 4 5894 0 616 0 0018 0 029 0 998 0 022 0 711 Table 17 Parameters of the marginal functions estimated using the FML Standard deviation of parameter estimates is given in parentheses Marginal function price Marginal function WP Mean SD DF Mean SD Skew 0 01239 0 9653 5 0206 0 00299 0 9993 0 6879
76. urring with non zero probability under the assumption of student t distribution The method used for correcting is as follows First the most extreme outliers are identified as the observations depart ing greatly from the 1 1 line in the QQ plots of the price residuals For DK1 the outliers include observation 359 sample minimum and 523 sample max imum and for DK2 observation 31 sample maximum 596 sample s second largest value and 359 sample minimum Using a block length of 15 days samples of block maxima and block minima are collected from the full 1080 day sample GEV distributions are fitted to the block maxima series and the series of additive inverse of block minima Based on the fitted GEV distributions the expected 1080 day return level is calculated and the maximum and minimum observations are replaced with the expected return levels The 1080 day return level is defined as the upper 1 1080 quantile of the residual distribution for maximum and lower 1 1080 quantile for the minimum For DK2 where also the second largest observation needs adjustment the second largest observation is replaced with the upper 2 1080 quantile that is the return level for 1080 2 540 days 40 DK1 QQ plot for residuals student t DK1 QQ plot for residuals adjusted tail student t o 4 oveytte 1 1 line sscerete 1 1 line a4 regression line w regression line P f e 3 95 confidence bands 3 95 confide
77. values of the copulas which all lie close to the values leading to independence None of the fitted copulas could be rejected by the S goodness of fit test This result could be explained by independent observations as copulas similar to the independence copula would not be rejected if observations are independent Moreover the extreme independence is supported by the y plots and x plots for the full sample which exhibit upper tail independence for both DK1 and DK2 Despite large estimate variation the chi plots of the block maxima indicate independence for the block maxima which is equivalent to upper tail independence While the results of the full sample copula modeling show positive depen dency between price and WP residuals results of the extreme value copula fitting indicate that the maxima occur independently 60 4 Conclusion In this chapter the results of the investigation of the dependence structure presented in Section 3 are summarized and conclusion in relation to the overall research questions is reached 4 1 Summary of results The results of the four steps of this study including price series filtering wind power series filtering modeling of full sample dependence and modeling of extreme dependence is presented below 1 The day ahead daily power prices are filtered using an ARMA GARCH model with weekday indicator variables under the assumption of t distributed residuals While the optimal order of the model
78. w to identify the case of indepen dence Likelihood ratio test Independence can be expressed in terms of the inde pendence copula as in equation 7 and if the independence copula is nested within the parametric copula a likelihood ratio test can be used to reject the hypothesis of independence However the likelihood ratio test requires that the parameter values which result in the independence copula are inside the boundaries and not on the boundary of the permitted parameter values Of the copulas presented in Table 1 and 2 only the normal copula lives up to this condition as the parameter value p 0 leads to independence and lie within the boundaries 1 lt p lt 1 Hence only for the normal copula the likelihood 22 ratio test can be used The likelihood ratio test statistic is defined as D 2 l lo 19 where J is the log likelihood of the model Mg with estimated parameters 6 and lo is the log likelihood of the independence copula Under the null hypoth esis that the observations are independent the test statistic will be approxi mately x p distributed The p degrees of freedom are equal to the difference in number of parameters in the two models however as the independence copula has no parameters p will be the number of parameters in Mg Coles 2001 Due of the limitations in the application of the likelihood ratio test a non parametric method using visual inspection is also employed y plot and
79. y a major role in the future power generation but integrating wind power into the electricity sys tem is not without challenges One aspect is the fact that the wind power generation capacity depends on an uncontrollable input the wind This is in clear contrast with classic fossil fuel based electricity production methods Wind power generation has low marginal cost thus electricity market prices are expected to decrease as wind power replaces power generated from gas and coal which has relatively high marginal production costs However as the wind power production is fluctuating with the energy level of the wind times of no wind are expected to lead to increased market prices and enhance the risk of price spikes High volatility and frequent occurrences of extremely high prices result in higher market risk and can lead to increased cost of financial risk management This thesis should be seen as a contribution to the debate about wind power as causing instability in electricity market The dependency between wind power production and the electricity prices in the Danish power market in which wind power contributes to more than 30 percent of the total elec tricity consumption is investigated both in term of the general relation and in connection to maximum prices Overall the following questions are to be adressed for the Danish day ahead electricity market e Is the electricity price and the magnitude of the wind power production related

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