Joint and conditional dependence modelling of peak district heating demand and outdoor temperature: a copula-based approach

A Correction to this article was published on 03 September 2019

This article has been updated

Abstract

This paper examines the complex dependence between peak district heating demand and outdoor temperature. Our aim is to provide the probability law of heat demand given extreme weather conditions, and derive useful implications for the management and production of thermal energy. We propose a copula-based approach and consider the case of the city of Bozen-Bolzano. The analysed data concern daily maxima heat demand observed from January 2014 to November 2017 and the corresponding outdoor temperature. We model the univariate marginal behaviour of the time series of heat demand and temperature with autoregressive integrated moving average models. Next, we investigate the dependence between the residuals’ time series through several copula models. The selected copula exhibits heavy-tailed and symmetric dependence. When taking into account the conditional behaviour of heat demand given extreme climatic events, the latter strongly affects the former, and we find a high probability of thermal energy demand reaching its peak.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Change history

  • 03 September 2019

    The original version of this article unfortunately contained a mistake.

References

  1. Aloui R, Ben Aissa M, Hammoudeh S, Nguyen D (2014) Dependence and extreme dependence of crude oil and natural gas prices with applications to risk management. Energy Econ 42C:332–342

    Article  Google Scholar 

  2. Ambach D, Croonenbroeck C (2016) Space-time short-to medium-term wind speed forecasting. Stat Methods Appl 25(1):5–20

    MathSciNet  MATH  Article  Google Scholar 

  3. Amjady N (2001) Short-term hourly load forecasting using time-series modeling with peak load estimation capability. IEEE Trans Power Syst 16(4):498–505

    Article  Google Scholar 

  4. Balic D, Gudmundsson O, Wilk V, Sokolov D, Iakimetc EE, Grundahl L, Duquette J, Razani AR, Hansen AB, Nastasi B, Vigants G, Kotenko M, Mohammadi S, Qvist K, Larsen DS, Hasberg K, Fritz S, Casetta D, Thellufsen JZ, Zhang L, Pan OMd, Hansen K, Gilski P, Bevilacqua C (2015) Smart energy systems and 4th generation district heating. Int J Sustain Energy Plan Manag 10:25–27

    Google Scholar 

  5. Ben Ghorbal N, Genest C, Nešlehová J (2009) On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence. Can J Stat 37(4):534–552

    MathSciNet  MATH  Article  Google Scholar 

  6. Box G, Cox D (1964) An analysis of transformations. J R Stat Soc B 26:211–243

    MATH  Google Scholar 

  7. Box G, Jenkins G (1970) Time series analysis: forecasting and control. Holden-Day, San Francisco

    MATH  Google Scholar 

  8. Brechmann E, Schepsmeier U (2013) Modeling dependence with C- and D-vine copulas: the R package CDVine. J Stat Softw 52(3):543–552

    Article  Google Scholar 

  9. Brunner MI, Sikorska AE, Seibert J (2018) Bivariate analysis of floods in climate impact assessments. Sci Total Environ 616–617:1392–1403

    Article  Google Scholar 

  10. Catalina T, Iordache V, Caracaleanu B (2013) Multiple regression model for fast prediction of the heating energy demand. Energy Build 57:302–312

    Article  Google Scholar 

  11. Charpentier A, Fermanian JD, Scaillet O (2007) The estimation of copulas: theory and practice. In: Jörn Rank (Ed) Copulas: from theory to application in finance. Risk Books, London, pp 35–64

  12. Cherubini U, Luciano E, Vecchiato W (2004) Copula methods in finance. Wiley, Chichester

    MATH  Book  Google Scholar 

  13. Connolly D, Mathiesen BV, Østergaard PA, Möller B, Nielsen S, Lund H, Trier D, Persson U, Nilsson D, Werner S (2012) Heat roadmap Europe 1: first pre-study for the EU27. Technical report

  14. Connolly D, Mathiesen BV, Østergaard PA, Nielsen S, Persson U, Werner S (2013) Heat roadmap Europe 2050: second pre-study for the EU27. Energy engineering

  15. Dahl M, Brun A, Andresen GB (2017) Using ensemble weather predictions in district heating operation and load forecasting. Appl Energy 193:455–465

    Article  Google Scholar 

  16. Dotzauer E (2002) Simple model for prediction of loads in district-heating systems. Appl Energy 73(3–4):277–284

    Article  Google Scholar 

  17. Duquette J, Rowe A, Wild P (2016) Thermal performance of a steady state physical pipe model for simulating district heating grids with variable flow. Appl Energy 178:383–393

    Article  Google Scholar 

  18. Durante F, Sempi C (2015) Principles of copula theory. CRC Press, Boca Raton

    MATH  Book  Google Scholar 

  19. Durante F, Fernández-Sánchez J, Pappadà R (2015) Copulas, diagonals, and tail dependence. Fuzzy Sets Syst 264(C):22–41

    MathSciNet  MATH  Article  Google Scholar 

  20. Durante F, Pappada R, Torelli N (2015b) Clustering of time series via non-parametric tail dependence estimation. Stat Pap 56(3):701–721

    MathSciNet  MATH  Article  Google Scholar 

  21. Ferraty F, Goia A, Salinelli E, Vieu P (2014) Peak-load forecasting using a functional semi-parametric approach. Top Nonparametr Stat 74:105–114

    MathSciNet  MATH  Article  Google Scholar 

  22. Genest C, Ghoudi K, Rivest LP (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82:543–552

    MathSciNet  MATH  Article  Google Scholar 

  23. Genest C, Rémillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44:199–213

    MathSciNet  MATH  Article  Google Scholar 

  24. Goia A, May C, Fusai G (2010) Functional clustering and linear regression for peak load forecasting. Int J Forecast 26(4):700–711

    Article  Google Scholar 

  25. Grønneberg S, Hjort NL (2014) The copula information criteria. Scand J Stat 41:436–459

    MathSciNet  MATH  Article  Google Scholar 

  26. Hofert M, Kojadinovic I, Maechler M, Yan J (2017) copula: multivariate dependence with copulas. R package version 0.999-18

  27. Howard B, Parshall L, Thompson J, Hammer S, Dickinson J, Modi V (2012) Spatial distribution of urban building energy consumption by end use. Energy Build 45:141–151

    Article  Google Scholar 

  28. Huang Z, Yu H, Peng Z, Zhao M (2015) Methods and tools for community energy planning: a review. Renew Sustain Energy Rev 42:1335–1348

    Article  Google Scholar 

  29. Hutchinson TP, Lai CD (1990) Continuous bivariate distributions, emphasising applications. Rumsby Scientific Publishing, Sydney

    MATH  Google Scholar 

  30. Jacobsen H (2000) Technology diffusion in energy-economy models: the case of danish vintage models. Energy J 21(1):43–71

    Google Scholar 

  31. Joe H (2011) Dependence modeling: vine copula handbook. In: Tail dependence in vine copulae. World scientific, Singapore, pp 165–187

  32. Kato K, Sakawa M, Ishimaru K, Ushiro S, Shibano T (2008) Heat load prediction through recurrent neural network in district heating and cooling systems. In: IEEE international conference on systems, man and cybernetics, 2008. SMC 2008. IEEE, pp 1401–1406

  33. Kazas G, Fabrizio E, Perino M (2017) Energy demand profile generation with detailed time resolution at an urban district scale: a reference building approach and case study. Appl Energy 193:243–262

    Article  Google Scholar 

  34. Kim JM, Jung H (2018) Dependence structure between oil prices, exchange rates, and interest rates. Energy J 39(2):233–258

    Article  Google Scholar 

  35. Kojadinovic I, Yan J (2010) Modeling multivariate distributions with continuous margins using the copula R package. J Stat Softw 34(9):1–20

    Article  Google Scholar 

  36. Kojadinovic I, Segers J, Yan J (2011) Large-sample tests of extreme-value dependence for multivariate copulas. Can J Stat 39(4):703–720

    MathSciNet  MATH  Article  Google Scholar 

  37. Kwiatkowski D, Phillips P, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. J Econom 54:159–178

    MATH  Article  Google Scholar 

  38. Lund H, Mathiesen BV (2009) Energy system analysis of 100% renewable energy systems—the case of Denmark in years 2030 and 2050. Energy 34(5):524–531

    Article  Google Scholar 

  39. Lund H, Möller B, Mathiesen BV, Dyrelund A (2010) The role of district heating in future renewable energy systems. Energy 35(3):1381–1390

    Article  Google Scholar 

  40. Lund H, Werner S, Wiltshire R, Svendsen S, Thorsen JE, Hvelplund F, Mathiesen BV (2014) 4th Generation District Heating (4GDH). Integrating smart thermal grids into future sustainable energy systems. Energy 68:1–11

    Article  Google Scholar 

  41. Ma W, Fang S, Liu G, Zhou R (2017) Modeling of district load forecasting for distributed energy system. Appl Energy 204:181–205

    Article  Google Scholar 

  42. Mathiesen BV, Connolly D, Lund H, Nielsen MP, Schaltz E, Wenzel H, Bentsen NS, Felby C, Kaspersen P, Ridjan I et al (2014) Ceesa 100% renewable energy transport scenarios towards 2050: technical background report part 2

  43. Nelsen RB (2006) Introduction to copulas. Springer, New York

    MATH  Google Scholar 

  44. Patton A (2012) A review of copula models for economic time series. J Multivar Anal 110:4–18

    MathSciNet  MATH  Article  Google Scholar 

  45. Persson U (2015) Current and future prospects for heat recovery from waste in European district heating systems: a literature and data review. Energy 110:25–26

    Google Scholar 

  46. Pircalabu A, Hvolby T, Jung J, Høg E (2017) Joint price and volumetric risk in wind power trading: a copula approach. Energy Econ 62(C):139–154

    Article  Google Scholar 

  47. Popescu D, Ungureanu F, Hernández-Guerrero A (2009) Simulation models for the analysis of space heat consumption of buildings. Energy 34(10):1447–1453

    Article  Google Scholar 

  48. Prando D, Patuzzi F, Pernigotto G, Gasparella A, Baratieri M (2014) Biomass gasification systems for residential application: an integrated simulation approach. Appl Therm Eng 71:152–160

    Article  Google Scholar 

  49. Protić M, Shamshirband S, Petković D, Abbasi A, Mat Kiah ML, Unar JA, Živković L, Raos M (2015) Forecasting of consumers heat load in district heating systems using the support vector machine with a discrete wavelet transform algorithm. Energy 87:343–351

    Article  Google Scholar 

  50. Raza MQ, Khosravi A (2015) A review on artificial intelligence based load demand forecasting techniques for smart grid and buildings. Renew Sustain Energy Rev 50:1352–1372

    Article  Google Scholar 

  51. Reinhart CF, Cerezo Davila C (2016) Urban building energy modeling—a review of a nascent field. Build Environ 97:196–202

    Article  Google Scholar 

  52. Rémillard B, Scaillet O (2009) Testing for equality between two copulas. J Multivar Anal 100(3):377–386

    MathSciNet  MATH  Article  Google Scholar 

  53. Rezaie B, Rosen MA (2012) District heating and cooling: review of technology and potential enhancements. Appl Energy 93:2–10

    Article  Google Scholar 

  54. Said S, Dickey D (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71:599–607

    MathSciNet  MATH  Article  Google Scholar 

  55. Salazar Y, Ng W (2015) Nonparametric estimation of general multivariate tail dependence and applications to financial time series. Stat Methods Appl 24:121–158

    MathSciNet  MATH  Article  Google Scholar 

  56. Serinaldi F, Bárdossy A, Kilsby C (2015) Upper tail dependence in rainfall extremes: would we know it if we saw it? Stoch Environ Res Risk Assess 29(4):1211–1233

    Article  Google Scholar 

  57. Sklar A (1959) Fonctions de répartition à \(n\) dimensions et leures marges. Publications de l’Institut de Statistique de L’Université de Paris 8:229–231

    MathSciNet  MATH  Google Scholar 

  58. Smith M, Hargroves K, Stasinopoulos P, Stephens R, Desha C (2007) Energy transformed: sustainable energy solutions for climate change mitigation. The Natural Edge Project, CSIRO, and Griffith University, Australia. https://eprints.qut.edu.au/85180/

  59. Suganthi L, Samuel AA (2012) Energy models for demand forecasting—a review. Renew Sustain Energy Rev 16:1223–1240

    Article  Google Scholar 

  60. Swan LG, Ugursal VI (2009) Modeling of end-use energy consumption in the residential sector: a review of modeling techniques. Renew Sustain Energy Rev 13:1819–1835

    Article  Google Scholar 

  61. Trivedi PK, Zimmer DM (2007) Copula modeling: an introduction for practitioners. Found Trends Econom 1(1):1–111

    MATH  Article  Google Scholar 

  62. Verrilli F, Srinivasan S, Gambino G, Canelli M, Himanka M, Vecchio CD, Sasso M, Glielmo L (2016) Model predictive control-based optimal operations of district heating system with thermal energy storage and flexible loads. IEEE Trans Autom Sci Eng 14(2):547–557

    Article  Google Scholar 

  63. Wang X, Smith K, Hyndman R (2006) Characteristic-based clustering for time series data. Data Min Knowl Discov 13(3):335–364

    MathSciNet  Article  Google Scholar 

  64. Wang H, Wang H, Zhou H, Zhu T (2018) Modeling and optimization for hydraulic performance design in multi-source district heating with fluctuating renewables. Energy Convers Manag 156:113–129

    Article  Google Scholar 

  65. Wojdyga K (2008) An influence of weather conditions on heat demand in district heating systems. Energy Build 40(11):2009–2014

    Article  Google Scholar 

  66. Wu L, Kaiser G, Solomon D, Winter R, Boulanger A, Anderson R (2012) Improving efficiency and reliability of building systems using machine learning and automated online evaluation. In: 2012 IEEE Long Island, systems, applications and technology conference (LISAT). IEEE, pp 1–6

  67. Yan J (2007) Enjoy the joy of copulas: with a package copula. J Stat Softw 21(4):1–21

    Article  Google Scholar 

  68. Zhang S, Okhrin O, Zhou Q, Song PK (2016) Goodness-of-fit test for specification of semiparametric copula dependence models. J Econ 193(1):215–233

    MathSciNet  MATH  Article  Google Scholar 

  69. Zimmer D, Trivedi P (2006) Using trivariate copulas to model sample selection and treatment effects: application to family health care demand. J Bus Econ Stat 24:63–76

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to two referees for the many useful suggestions that have helped improve this paper. The first author (corresponding author F. Marta L. Di Lascio) acknowledges the support of the Free University of Bozen-Bolzano, Faculty of Economics and Management, via the project “The use of Copula for the Analysis of Complex and Extreme Energy and Climate data (CACEEC)” (Grant Nos. WW200S). The second author (Andrea Menapace) acknowledges Alperia and the Bozen-Bolzano province for providing the analysed data. The third author (Maurizio Righetti) acknowledges the support of the Free University of Bozen-Bolzano via the interdisciplinary project “Methods for optimization and integration given energy prices and renewable resources forecasts (MOIEREF)” (Grant Nos. WW2096).

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. Marta L. Di Lascio.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: The original version of this article unfortunately contained a mistake. Maurizio Righetti’s given name and family name had been presented in the wrong order. The correct order is given here.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Di Lascio, F.M.L., Menapace, A. & Righetti, M. Joint and conditional dependence modelling of peak district heating demand and outdoor temperature: a copula-based approach. Stat Methods Appl 29, 373–395 (2020). https://doi.org/10.1007/s10260-019-00488-4

Download citation

Keywords

  • Copula function
  • Conditional probability
  • District heating system
  • Outdoor temperature
  • Peak heat demand
  • SARIMA models