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Are multifractal processes suited to forecasting electricity price volatility? Evidence from Australian intraday data

Mawuli Segnon, Chi Keung Lau, Bernd Wilfling and Rangan Gupta

Abstract

We analyze Australian electricity price returns and find that they exhibit volatility clustering, long memory, structural breaks, and multifractality. Consequently, we let the return mean equation follow two alternative specifications, namely (i) a smooth transition autoregressive fractionally integrated moving average (STARFIMA) process, and (ii) a Markov-switching autoregressive fractionally integrated moving average (MSARFIMA) process. We specify volatility dynamics via a set of (i) short- and long-memory GARCH-type processes, (ii) Markov-switching (MS) GARCH-type processes, and (iii) a Markov-switching multifractal (MSM) process. Based on equal and superior predictive ability tests (using MSE and MAE loss functions), we compare the out-of-sample relative forecasting performance of the models. We find that the (multifractal) MSM volatility model keeps up with the conventional GARCH- and MSGARCH-type specifications. In particular, the MSM model outperforms the alternative specifications, when using the daily squared return as a proxy for latent volatility.

JEL Classification: C22; C52; C53

Corresponding author: Bernd Wilfling, Westfälische Wilhelms-Universität, Münster, Germany, E-mail:

Acknowledgments

We are grateful to Bruce Mizrach, Christian Hafner, and two reviewers for their constructive comments, which greatly improved the paper. The usual disclaimer applies.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

Aggarwal, S. K., L. M. Saini, and A. Kumar. 2009. “Electricity Price Forecasting in Deregulated Markets: A Review and Evaluation.” International Journal of Electrical Power & Energy Systems 31: 13–22, https://doi.org/10.1016/j.ijepes.2008.09.003.Search in Google Scholar

Amjady, N. 2006. “Day-Ahead Price Forecasting of Electricity Markets by New Fuzzy Neural Network.” IEEE Transactions on Power Systems 21: 887–996, https://doi.org/10.1109/tpwrs.2006.873409.Search in Google Scholar

Amjady, N. 2012. “Short-Term Electricity Price Forecasting.” In Electric Power Systems: Advanced Forecasting Techniques and Optimal Generation Scheduling, edited by J. P. S. Catalão, 1–58. Boca Raton: CRC Press. Chapter 4.Search in Google Scholar

Amjady, N., and M. Hemmati. 2009. “Day-Ahead Price Forecasting of Electricity Markets by a Hybrid Intelligent System.” European Transactions on Electrical Power 19: 89–102, https://doi.org/10.1002/etep.242.Search in Google Scholar

Andersen, T. G., and T. Bollerslev. 1998. “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts.” International Economic Review 39: 885–905, https://doi.org/10.2307/2527343.Search in Google Scholar

Apergis, N., and M. C. K. Lau. 2015. “Structural Breaks and Electricity Prices: Further Evidence on the Role of Climate Policy Uncertainties in the Australian Electricity Market.” Energy Economics 52: 176–82, https://doi.org/10.1016/j.eneco.2015.10.014.Search in Google Scholar

Awartani, B. M. A., and V. Corradi. 2005. “Predicting the Volatility of the S&P 500 Stock Index via GARCH Models: The Role of Asymmetries.” International Journal of Forecasting 21: 167–83, https://doi.org/10.1016/j.ijforecast.2004.08.003.Search in Google Scholar

Baillie, R. T., T. Bollerslev, and H. O. Mikkelsen. 1996. “Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 74: 3–30, https://doi.org/10.1016/s0304-4076(95)01749-6.Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–27, https://doi.org/10.1016/0304-4076(86)90063-1.Search in Google Scholar

Bollerslev, T., R. F. Engle, and D. Nelson. 1994. “ARCH Models.” In Handbook of Econometrics, edited by R. F. Engle and D. L. McFadden, 2959–3038. Amsterdam: Elsevier Science.Search in Google Scholar

Bougerol, P., and N. Picard. 1992. “Stationarity of GARCH Processes and of Some Nonnegative Time Series.” Journal of Econometrics 52: 115–27, https://doi.org/10.1016/0304-4076(92)90067-2.Search in Google Scholar

Box, G. E. P., and D. R. Cox. 1964. “An Analysis of Transformations.” Journal of the Royal Statistical Society B26: 211–52, https://doi.org/10.1111/j.2517-6161.1964.tb00553.x.Search in Google Scholar

Calvet, L., and A. Fisher. 2001. “Forecasting Multifractal Volatility.” Journal of Econometrics 105: 27–58, https://doi.org/10.1016/s0304-4076(01)00069-0.Search in Google Scholar

Calvet, L., and A. Fisher. 2004. “Regime-Switching and the Estimation of Multifractal Processes.” Journal of Financial Econometrics 2: 44–83, https://doi.org/10.1093/jjfinec/nbh003.Search in Google Scholar

Catalão, J. P. S., S. J. P. S. Mariano, V. M. F. Mendes, and L. A. F. M. Ferreria. 2007. “Short-Term Electricity Prices Forecasting in a Competitive Market: A Neural Network Approach.” Electric Power Systems Research 77: 1297–304, https://doi.org/10.1016/j.epsr.2006.09.022.Search in Google Scholar

Chan, K. F., and P. Gray. 2006. “Using Extreme Value Theory to Measure Value-At-Risk for Daily Electricity Spot Prices.” International Journal of Forecasting 22: 283–300.Search in Google Scholar

Chan, K. F., P. Gray, and B. V. Campen. 2008. “A New Approach to Characterizing and Forecasting Electricity Price Volatility.” International Journal of Forecasting 24: 728–43, https://doi.org/10.1016/j.ijforecast.2008.08.002.Search in Google Scholar

Christensen, T. M., A. S. Hurn, and K. A. Lindsay. 2012. “Forecasting Spikes in Electricity Prices.” International Journal of Forecasting 28: 400–11, https://doi.org/10.1016/j.ijforecast.2011.02.019.Search in Google Scholar

Cifter, A. 2013. “Forecasting Electricity Price Volatility with Markov-Switching GARCH Model: Evidence from the Nordic Electric Power Market.” Electric Power Systems Research 102: 61–7, https://doi.org/10.1016/j.epsr.2013.04.007.Search in Google Scholar

Clements, A. E., R. Herrera, and A. S. Hurn. 2015. “Modelling Interregional Links in Electricity Price Spikes.” Energy Economics 51: 383–93, https://doi.org/10.1016/j.eneco.2015.07.014.Search in Google Scholar

Conrad, C., and B. R. Haag. 2006. “Inequality Constraints in the Fractionally Integrated GARCH Model.” Journal of Financial Econometrics 4: 413–49, https://doi.org/10.1093/jjfinec/nbj015.Search in Google Scholar

Contreras, J., R. Espinola, F. J. Nogales, and A. J. Conejo. 2003. “ARIMA Models to Predict Next-Day Electricity Prices.” IEEE Transactions on Power Systems 18: 1014–20, https://doi.org/10.1109/tpwrs.2002.804943.Search in Google Scholar

Diebold, F. X. 2015. “Comparing Predictive Accuracy, Twenty Years Later: A Personal Perspective on the Use and Abuse of Diebold-Mariano Tests.” Journal of Business and Economic Statistics 33: 1–10, https://doi.org/10.1080/07350015.2014.983236.Search in Google Scholar

Diebold, F. X., and R. S. Mariano. 1995. “Comparing Predictive Accuracy.” Journal of Business and Economic Statistics 13: 253–63, https://doi.org/10.2307/1392185.Search in Google Scholar

Ding, Z., C. Granger, and R. F. Engle. 1993. “A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance 1: 83–106. https://doi.org/10.1016/0927-5398(93)90006-d.Search in Google Scholar

Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50: 987–1007, https://doi.org/10.2307/1912773.Search in Google Scholar

Garcia, R. J., J. Contreras, M. V. Akkeren, and J. B. C. Garcia. 2005. “A GARCH Forecasting Model to Predict Day-Ahead Electricity Prices.” IEEE Transactions on Power Systems 20: 867–74, https://doi.org/10.1109/tpwrs.2005.846044.Search in Google Scholar

Garcia-Martos, R. C., J. Rodriguez, and M. J. Sanchez. 2007. “Mixed Models for Shortrun Forecasting of Electricity Prices: Application for the Spanish Market.” IEEE Transactions on Power Systems 22: 544–51, https://doi.org/10.1109/tpwrs.2007.894857.Search in Google Scholar

Gianfreda, A. 2010. “Volatility and Volume Effects in European Electricity Markets.” Economic Notes 1: 47–63, https://doi.org/10.1111/j.1468-0300.2009.00220.x.Search in Google Scholar

Gianfreda, A., and L. Grossi. 2012. Volatility Models for Electricity Prices with Intra-daily Information. Mimeo: University of Modena (UNIMORE). Available at https://ssrn.com/abstract=2188148.Search in Google Scholar

Glosten, L., R. Jagannathan, and D. E. Runkle. 1993. “On the Relation between the Expected Value and Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance 46: 1779–801, https://doi.org/10.1111/j.1540-6261.1993.tb05128.x.Search in Google Scholar

Goss, B. 2006. “Liquidity, Volume and Volatility in US Electricity Futures: The Case of Palo Verde.” Applied Financial Economics Letters 2: 43–6, https://doi.org/10.1080/17446540500396974.Search in Google Scholar

Haas, M., S. Mittnik, and M. S. Paolella. 2004. “A New Approach to Markov-Switching GARCH Models.” Journal of Financial Econometrics 2: 493–530, https://doi.org/10.1093/jjfinec/nbh020.Search in Google Scholar

Haldrup, N., and M. Ø. Nielsen. 2006. “A Regime Switching Long Memory Model for Electricity Prices.” Journal of Econometrics 135: 349–76, https://doi.org/10.1016/j.jeconom.2005.07.021.Search in Google Scholar

Hansen, P. R. 2005. “A Test for Superior Predictive Ability.” Journal of Business and Economic Statistics 23: 365–80, https://doi.org/10.1198/073500105000000063.Search in Google Scholar

Haugom, E., and C. J. Ullrich. 2012. “Forecasting Spot Price Volatility Using the Short-Term Forward Curve.” Energy Economics 34: 1826–33, https://doi.org/10.1016/j.eneco.2012.07.017.Search in Google Scholar

Haugom, E., S. Westgaard, P. B. Solibakke, and G. Lien. 2011. “Realized Volatility and the Influence of Market Measures on Predictability: Analysis of Nord Pool Forward Electricity Data.” Energy Economics 33: 1206–15, https://doi.org/10.1016/j.eneco.2011.01.013.Search in Google Scholar

He, C., and T. Teräsvirta. 1999. “Properties of Moments of a Family of GARCH Processes.” Journal of Econometrics 92: 173–92. https://doi.org/10.1016/s0304-4076(98)00089-x.Search in Google Scholar

Hentschel, L. 1995. “All in the Family: Nesting Symmetric and Asymmetric GARCH Models.” Journal of Financial Economics 39: 71–104. https://doi.org/10.1016/0304-405x(94)00821-h.Search in Google Scholar

Higgs, H., and A. C. Worthington. 2005. “Systematic Features of High-Frequency Volatility in Australian Electricity Markets: Intraday Patterns, Information Arrival and Calender Effects.” The Energy Journal 26: 23–40. https://doi.org/10.5547/issn0195-6574-ej-vol26-no4-2.Search in Google Scholar

Higgs, H., and A. C. Worthington. 2008. “Stochastic Price Modeling of High Volatility, Mean-Reverting, Spike-Prone Commodities: The Australian Wholesale Spot Electricity Market.” Energy Economics 30: 3172–85, https://doi.org/10.1016/j.eneco.2008.04.006.Search in Google Scholar

Hillebrand, E., and M. C. Medeiros. 2016. “Nonlinearity, Breaks, and Long-Range Dependence in Time-Series Models.” Journal of Business and Economic Statistics 34: 23–41, https://doi.org/10.1080/07350015.2014.985828.Search in Google Scholar

Hong, T. 2015. “Energy Forecasting: Past, Present, and Future.” Foresight: The International Journal of Applied Forecasting 32: 43–8.Search in Google Scholar

Huisman, R., and R. Mahieu. 2003. “Regime Jumps in Electricity Prices.” Energy Economics 25: 425–34, https://doi.org/10.1016/s0140-9883(03)00041-0.Search in Google Scholar

Huurman, C., F. Ravazzolo, and C. Zhou. 2012. “The Power of Weather.” Computational Statistics 56: 3793–807, https://doi.org/10.1016/j.csda.2010.06.021.Search in Google Scholar

Jonsson, E., and R. E. Dahl. 2016. Regime Shifts in Electricity Prices in USA and EU. . Mimeo: University of Stavanger. Available at https://www.iaee.org/proceedings/article/13883.Search in Google Scholar

Kantelhardt, J. W., S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley. 2002. “Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series.” Physica A316: 87–114, https://doi.org/10.1016/s0378-4371(02)01383-3.Search in Google Scholar

Karakatsani, N. V., and D. W. Bunn. 2008. “Modelling the Volatility of Spot Electricity Prices.” International Journal of Forecasting 24: 764–85, https://doi.org/10.1016/j.ijforecast.2008.09.008.Search in Google Scholar

Kiliç, R. 2011. “Long Memory and Nonlinearity in Conditional Variances: A Smooth Transition FIGARCH Model.” Journal of Empirical Finance 18: 368–78, https://doi.org/10.1016/j.jempfin.2011.07.004.Search in Google Scholar

Knittel, C. R., and M. R. Roberts. 2005. “An Empirical Examination of Restructured Electricity Prices.” Energy Economics 27: 791–817, https://doi.org/10.1016/j.eneco.2004.11.005.Search in Google Scholar

Koopman, S. J., M. Ooms, and M. A. Carnero. 2007. “Periodic Seasonal Reg-ARFIMAGARCH Models for Daily Electricity Spot Prices.” Journal of the American Statistical Association 102: 16–27, https://doi.org/10.1198/016214506000001022.Search in Google Scholar

Kristiansen, T. 2012. “Forecasting Nord Pool Day-Ahead Prices with an Autoregressive Model.” Energy Policy 49: 328–32, https://doi.org/10.1016/j.enpol.2012.06.028.Search in Google Scholar

Lago, J., F. De Ridder, and B. De Schutter. 2018. “Forecasting Spot Electricity Prices: Deep Learning Approaches and Empirical Comparison of Traditional Algorithms.” Applied Energy 221: 386–405, https://doi.org/10.1016/j.apenergy.2018.02.069.Search in Google Scholar

Ling, S., and M. McAleer. 2002a. “Necessary and Sufficient Moment Conditions for the GARCH(r,s) and Asymmetric Power GARCH(r,s) Models.” Econometric Theory 18: 722–9, https://doi.org/10.1017/s0266466602183071.Search in Google Scholar

Ling, S., and M. McAleer. 2002b. “Stationarity and the Existence of Moments of a Family of GARCH Processes.” Journal of Econometrics 106: 109–17, https://doi.org/10.1016/s0304-4076(01)00090-2.Search in Google Scholar

Liu, J.-C. 2006. “Stationarity of a Markov-Switching GARCH Model.” Journal of Financial Econometrics 4: 573–93, https://doi.org/10.1093/jjfinec/nbl004.Search in Google Scholar

Liu, R., T. di Matteo, and T. Lux. 2007. “True and Apparent Scaling: The Proximity of the Markov-Switching Multifractal Model to Long-Range Dependence.” Physica A383: 35–42, https://doi.org/10.1016/j.physa.2007.04.085.Search in Google Scholar

Lux, T. 2008. “The Markov-Switching Multifractal Model of Asset Returns: GMM Estimation and Linear Forecasting of Volatility.” Journal of Business and Economic Statistics 26: 194–210, https://doi.org/10.1198/073500107000000403.Search in Google Scholar

Lux, T., and M. Segnon. 2018. “Multifractal Models in Finance: Their Origin, Properties and Applications.” In The Oxford Handbook on Computational Economics and Finance, edited by S.-H. Chen, M. Kaboudan, and Y.-R. Du, 204–48. New York: Oxford University Press.Search in Google Scholar

Lux, T., M. Segnon, and R. Gupta. 2016. “Forecasting Crude Oil Price Volatility and Value-At-Risk: Evidence from Historical and Recent Data.” Energy Economics 56: 117–33, https://doi.org/10.1016/j.eneco.2016.03.008.Search in Google Scholar

Maciejowska, K., J. Nowotarski, and R. Weron. 2016. “Probabilistic Forecasting of Electricity Spot Prices Using Factor Quantile Regression Averaging.” International Journal of Forecasting 32: 957–65, https://doi.org/10.1016/j.ijforecast.2014.12.004.Search in Google Scholar

Mandelbrot, B. B. 1974. “Intermittent Turbulence in Self Similar Cascades; Divergence of High Moments and Dimension of the Carrier.” Journal of Fluid Mechanics 62: 331–58, https://doi.org/10.1017/s0022112074000711.Search in Google Scholar

Mandelbrot, B. B., A. Fisher, and L. Calvet. 1997. “A Multifractal Model of Asset Returns.” In Cowles Foundation for Research in Economics. Yale University, Discussion Paper No. 1164.Search in Google Scholar

Marcucci, J. 2005. “Forecasting Stock Market Volatility with Regime-Switching GARCH Models.” Studies in Nonlinear Dynamics & Econometrics 9: 1–53, https://doi.org/10.2202/1558-3708.1145.Search in Google Scholar

McAleer, M., and M. C. Medeiros. 2008. “Realized Volatility: A Review.” Econometric Reviews 27: 10–45, https://doi.org/10.1080/07474930701853509.Search in Google Scholar

Misiorek, A., S. Trueck, and R. Weron. 2006. “Point and Interval Forecasting of Spot Electricity Prices: Linear vs. Non-linear Time Series Models.” Studies in Nonlinear Dynamics & Econometrics 10 (3):Article 2. https://doi.org/10.2202/1558-3708.1362.Search in Google Scholar

Möst, D., and D. Keles. 2010. “A Survey of Stochastic Modelling Approaches for Liberalized Electricity Markets.” European Journal of Operational Research 207: 543–56, https://doi.org/10.1016/j.ejor.2009.11.007.Search in Google Scholar

Nelson, D. B. 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica 59: 347–70, https://doi.org/10.2307/2938260.Search in Google Scholar

Nowotarski, J., E. Raviv, S. Trueck, and R. Weron. 2014. “An Empirical Comparison of Alternative Schemes for Combining Electricity Spot Price Forecasts.” Energy Economics 30: 1116–57, https://doi.org/10.1007/s00180-014-0523-0.Search in Google Scholar

Nowotarski, J., and R. Weron. 2015. “Computing Electricity Spot Price Prediction Intervals Using Quantile Regression and Forecast Averaging.” Computational Statistics 30: 791–803, https://doi.org/10.1007/s00180-014-0523-0.Search in Google Scholar

Patton, A. J. 2011. “Volatility Forecast Comparison Using Imperfect Volatility Proxies.” Journal of Econometrics 160: 246–56, https://doi.org/10.1016/j.jeconom.2010.03.034.Search in Google Scholar

Qu, H., W. Chen, M. Niu, and X. Li. 2016. “Forecasting Realized Volatility in Electricity Markets Using Logistic Smooth Transiton Heterogeneous Autoregressive Models.” Energy Economics 54: 68–76, https://doi.org/10.1016/j.eneco.2015.12.001.Search in Google Scholar

Raviv, E., K. E. Bouwman, and D. van Dijk. 2015. “Forecasting Day-Ahead Electricity Prices: Utilizing Hourly Prices.” Energy Economics 50: 227–39, https://doi.org/10.1016/j.eneco.2015.05.014.Search in Google Scholar

Reher, G., and B. Wilfling. 2016. “A Nesting Framework for Markov-Switching GARCH Modelling with an Application to the German Stock Market.” Quantitative Finance 16: 411–26, https://doi.org/10.1080/14697688.2015.1015599.Search in Google Scholar

Resnick, S., and C. Stǎricǎ. 1995. “Consistency of Hill’s Estimator for Dependent Data.” Journal of Applied Probability 32: 139–67, https://doi.org/10.2307/3214926.Search in Google Scholar

Resnick, S., and C. Stǎricǎ. 1997. “Asymptotic Behavior of Hill’s Estimator for Autoregressive Data.” Stochastic Models 13: 703–23, https://doi.org/10.1080/15326349708807448.Search in Google Scholar

Resnick, S., and C. Stǎricǎ. 1998. “Tail Index Estimation for Dependent Data.” The Annals of Applied Probability 8: 1156–83, https://doi.org/10.1214/aoap/1028903376.Search in Google Scholar

Sansó, A., V. Arragó, and J. L. Carrion. 2004. “Testing for Change in the Unconditional Variance of Financial Time Series.” Revista de Economiá Financiera 4: 32–53.Search in Google Scholar

Segnon, M., T. Lux, and R. Gupta. 2017. “Modeling and Forecasting the Volatility of Carbon Dioxide Emission Allowance Prices: A Review and Comparison of Modern Volatility Models.” Renewable and Sustainable Energy Reviews 69: 692–704, https://doi.org/10.1016/j.rser.2016.11.060.Search in Google Scholar

Swider, D. J., and C. Weber. 2007. “Extended ARMA Models for Estimating Price Developments on Day-Ahead Electricity Markets.” Electric Power Systems Research 77: 583–93, https://doi.org/10.1016/j.epsr.2006.05.013.Search in Google Scholar

Taylor, S. J. 1986. Modelling Financial Time Series. Chichester: John Wiley & Sons.Search in Google Scholar

Thompson, J. R., and J. R. Wilson. 2016. “Multifractal Detrended Fluctuation Analysis: Practical Applications to Financial Time Series.” Mathematics and Computers in Simulation 126: 63–88, https://doi.org/10.1016/j.matcom.2016.03.003.Search in Google Scholar

Tsay, W.-J., and W. K. Härdle. 2009. “A Generalized ARFIMA Process with Markov-Switching Fractional Differencing Parameter.” Journal of Statistical Computation and Simulation 79: 731–45, https://doi.org/10.1080/00949650801910239.Search in Google Scholar

Tse, Y. K. 1998. “The Conditional Heteroscedasticity of the Yen-Dollar Exchange Rate.” Journal of Applied Econometrics 13: 49–55. https://doi.org/10.1002/(sici)1099-1255(199801/02)13:1<49::aid-jae459>3.0.co;2-o.Search in Google Scholar

Vahidinasab, V., and S. J. Kazemi. 2008. “Day-Ahead Forecasting in Restructured Power Systems Using Artificial Neural Networks.” Electric Power Systems Research 78: 1332–42, https://doi.org/10.1016/j.epsr.2007.12.001.Search in Google Scholar

Wang, Y., C. Wu, and L. Yang. 2016. “Forecasting Crude Oil Market Volatility: A Markov Switching Multifractal Volatility Approach.” International Journal of Forecasting 32: 1–9, https://doi.org/10.1016/j.ijforecast.2015.02.006.Search in Google Scholar

Weron, R. 2002. “Estimating Long-Range Dependence: Finite Sample Properties and Confidence Intervals.” Physica A: Statistical Mechanics and its Applications 312: 285–99, https://doi.org/10.1016/s0378-4371(02)00961-5.Search in Google Scholar

Weron, R. 2014. “Electricity Price Forecasting: A Review of the State-Of-The-Art with Look into the Future.” International Journal of Forecasting 30: 1030–81, https://doi.org/10.1016/j.ijforecast.2014.08.008.Search in Google Scholar

Wu, L., and M. Shahidehpour. 2010. “A Hybrid Model for Day-Ahead Price Forecasting.” IEEE Transactions on Power Systems 25: 1519–30.Search in Google Scholar

Zareipour, H. 2012. “Short-term Electricity Market Prices: A Review of Characteristics and Forecasting Methods.” In Handbook of Networks in Power Systems I, edited by A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis and M. V. F. Pereira, 89–121. Heidelberg: Springer-Verlag.Search in Google Scholar


Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2019-0009).


Received: 2019-02-14
Accepted: 2020-10-30
Published Online: 2020-11-17

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