# Stochastic Generation of District Heat Load

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Physics-based models, instead, guarantee accurate heat load prediction, but with a huge amount of data about buildings and environment, and high computational cost. However, some simplification techniques, such as prototypical buildings methods, e.g., [19] and sample methods, e.g., [20], allow the application at the district scale;
- On the other hand, the spread of smart meters installed supporting the intelligent grids are able to provide high quality and frequency consumption data, enabling the significant growth of data-driven methods. These methods offer several alternative techniques for heat load modelling: regression-based models (e.g., multivariate linear regression model [21] and conditional demand analysis [22]), time series models (e.g., time-domain approaches such as ARX [23] and ARIMAX [24], and frequency domain approaches such as Klaman filter [25] and Fourier series [26]), intelligent models (e.g., artificial neural networks [27], support vector machines [28], and feature fusion LSTM [29]), and hybrid statistical models [30].

## 2. Methods

#### 2.1. Seasonal Dependence Modelling between Heat Demand and Outside Temperature

#### 2.2. Seasonal Dependence Modelling between Heat Demand and Outside Temperature

## 3. Case Study

#### 3.1. Subsection District Heating System of Bozen-Bolzano

#### 3.2. Dataset

## 4. Calculation

#### 4.1. Subsection

#### 4.2. Estimation of the Aggregation Relationship

^{2}/year, while the heating surface is the nominal area of the dwelling. These two values are easily obtainable: the heating surface is available to municipalities for tax reasons, and the energy class is required for house energy rating. However, other, more accurate ways of deriving P are allowed in this procedure, e.g., the direct use of the yearly metered data.

#### 4.3. Selection of the Distribution of Daily Pattern

#### 4.4. Linear Regression of Seasonal Trend

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | Units | Definition |

$d$ | day | day of the selected period |

$D$ | (-) | total number of the day of the selected period |

${E}_{c}$ | kWh/${\mathrm{m}}^{2}$/year | annual heat demand per unit of heating surface |

${G}_{a}$ | (-) | annual relative daily variation index |

${G}_{w}$ | (-) | annual relative seasonal variation index |

$HS$ | ${\mathrm{m}}^{2}$ | heating surface |

$n$ | (-) | quarter-hour time interval of a day |

$N$ | (-) | total number of the daily time step |

$P$ | kWh/year | total annual heat demand |

$T$ | °C | mean daily outdoor temperature |

${w}_{d}$ | (-) | daily heat load pattern |

${w}_{s}$ | (-) | seasonal heat load pattern |

${W}_{d,n}$ | kW | thermal power demand |

$\overline{W}$ | kW | annual average thermal power demand |

${W}_{d}$ | kW | daily average thermal power demand |

## References

- Mathiesen, B.V.; Lund, H.; Karlsson, K.B. 100% Renewable energy systems, climate mitigation and economic growth. Appl. Energy
**2011**, 88, 488–501. [Google Scholar] [CrossRef] - Fankhauser, S. Valuing Climate Change: The Economics of the Greenhouse; Routledge: London, UK, 2013. [Google Scholar]
- Mazhar, A.R.; Liu, S.; Shukla, A. A state of art review on the district heating systems. Renew. Sustain. Energy Rev.
**2018**, 96, 420–439. [Google Scholar] [CrossRef] - Lund, H.; Østergaard, P.A.; Chang, M.; Werner, S.; Svendsen, S.; Sorknæs, P.; Thorsen, J.E.; Hvelplund, F.; Mortensen, B.O.G.; Mathiesen, B.V.; et al. The status of 4th generation district heating: Research and results. Energy
**2018**, 164, 147–159. [Google Scholar] [CrossRef] - Lund, H. Renewable Energy Systems: A Smart Energy Systems Approach to the Choice and Modeling of 100% Renewable Solutions; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Rémillard, B.; Scaillet, O. Testing for equality between two copulas. J. Multivar. Anal.
**2008**, 100, 377–386. [Google Scholar] [CrossRef] [Green Version] - Guelpa, E.; Toro, C.; Sciacovelli, A.; Melli, R.; Sciubba, E.; Verda, V. Optimal operation of large district heating networks through fast fluid-dynamic simulation. Energy
**2016**, 102, 586–595. [Google Scholar] [CrossRef] - Righetti, M.; Bort, C.M.G.; Bottazzi, M.; Menapace, A.; Zanfei, A. Optimal Selection and Monitoring of Nodes Aimed at Supporting Leakages Identification in WDS. Water
**2019**, 11, 629. [Google Scholar] [CrossRef] [Green Version] - Suganthi, L.; Samuel, A.A. Energy models for demand forecasting—A review. Renew. Sustain. Energy Rev.
**2012**, 16, 1223–1240. [Google Scholar] - Wang, Z.; Srinivasan, R. A review of artificial intelligence based building energy use prediction: Contrasting the capabilities of single and ensemble prediction models. Renew. Sustain. Energy Rev.
**2016**, 75, 796–808. [Google Scholar] [CrossRef] - Ma, W.; Fang, S.; Liu, G.; Zhou, R. Modeling of district load forecasting for distributed energy system. Appl. Energy
**2017**, 204, 181–205. [Google Scholar] [CrossRef] - Lake, A.; Rezaie, B.; Beyerlein, S. Review of district heating and cooling systems for a sustainable future. Renew. Sustain. Energy Rev.
**2017**, 67, 417–425. [Google Scholar] [CrossRef] - Sun, Q.; Li, H.; Ma, Z.; Wang, C.; Campillo, J.; Zhang, Q.; Wallin, F.; Guo, J. A Comprehensive Review of Smart Energy Meters in Intelligent Energy Networks. IEEE Internet Things J.
**2015**, 3, 464–479. [Google Scholar] [CrossRef] - Uihlein, A.; Eder, P. Policy options towards an energy efficient residential building stock in the EU-27. Energy Build.
**2009**, 42, 791–798. [Google Scholar] [CrossRef] - Jaggs, M.; Palmer, J. Energy performance indoor environmental quality retrofit—A European diagnosis and decision making method for building refurbishment. Energy Build.
**2000**, 31, 97–101. [Google Scholar] [CrossRef] - Sailor, D.J.; Lu, L. A top–down methodology for developing diurnal and seasonal anthropogenic heating profiles for urban areas. Atmos. Environ.
**2004**, 38, 2737–2748. [Google Scholar] [CrossRef] - Siwei, L.; Huang, Y.J. Energy conservation standard for space heating in Chinese urban residential buildings. Energy
**1993**, 18, 871–892. [Google Scholar] [CrossRef] - Asare-Bediako, B.; Kling, W.; Ribeiro, P. Future residential load profiles: Scenario-based analysis of high penetration of heavy loads and distributed generation. Energy Build.
**2014**, 75, 228–238. [Google Scholar] [CrossRef] - Pedersen, L.; Stang, J.; Ulseth, R. Load prediction method for heat and electricity demand in buildings for the purpose of planning for mixed energy distribution systems. Energy Build.
**2008**, 40, 1124–1134. [Google Scholar] [CrossRef] - Ghiassi, N.; Hammerberg, K.; Taheri, M. An enhanced sampling-based approach to urban energy modelling. In Proceedings of the BS 2015—14th International IBPSA Conference, Hyderabad, India, 7–9 December 2015; pp. 626–632. [Google Scholar]
- Howard, B.; Parshall, L.; Thompson, J.; Hammer, S.; Dickinson, J.; Modi, V. Spatial distribution of urban building energy consumption by end use. Energy Build.
**2011**, 45, 141–151. [Google Scholar] [CrossRef] - Newsham, G.R.; Donnelly, C.L. A model of residential energy end-use in Canada: Using conditional demand analysis to suggest policy options for community energy planners. Energy Policy
**2013**, 59, 133–142. [Google Scholar] [CrossRef] [Green Version] - Sarwar, R.; Cho, H.; Cox, S.J.; Mago, P.J.; Luck, R. Field validation study of a time and temperature indexed autoregressive with exogenous (ARX) model for building thermal load prediction. Energy
**2017**, 119, 483–496. [Google Scholar] [CrossRef] - Powell, K.M.; Sriprasad, A.; Cole, W.J.; Edgar, T.F. Heating, cooling, and electrical load forecasting for a large-scale district energy system. Energy
**2014**, 74, 877–885. [Google Scholar] [CrossRef] - Gu, W.; Wang, Z.; Wu, Z.; Luo, Z.; Tang, Y.; Wang, J. An Online Optimal Dispatch Schedule for CCHP Microgrids Based on Model Predictive Control. IEEE Trans. Smart Grid
**2016**, 8, 2332–2342. [Google Scholar] [CrossRef] - Tsai, C.-L.; Chen, W.T.; Chang, C.-S. Polynomial-Fourier series model for analyzing and predicting electricity consumption in buildings. Energy Build.
**2016**, 127, 301–312. [Google Scholar] [CrossRef] - Kuo, P.-H.; Huang, C.-J. A High Precision Artificial Neural Networks Model for Short-Term Energy Load Forecasting. Energies
**2018**, 11, 213. [Google Scholar] [CrossRef] [Green Version] - Protić, M.; Shamshirband, S.; Petković, D.; Abbasi, A.; Mat Kiah, M.L.; Unar, J.A.; Živković, L.; Raos, M. Forecasting of consumers heat load in district heating systems using the support vector machine with a discrete wavelet transform algorithm. Energy
**2015**, 87, 343–351. [Google Scholar] [CrossRef] - Xue, G.; Pan, Y.; Lin, T.; Song, J.; Qi, C.; Wang, Z. District Heating Load Prediction Algorithm Based on Feature Fusion LSTM Model. Energies
**2019**, 12, 2122. [Google Scholar] [CrossRef] [Green Version] - Labidi, M.; Eynard, J.; Faugeroux, O.; Grieu, S. A new strategy based on power demand forecasting to the management of multi-energy district boilers equipped with hot water tanks. Appl. Therm. Eng.
**2017**, 113, 1366–1380. [Google Scholar] [CrossRef] - Reinhart, C.F.; Davila, C.C. Urban building energy modeling—A review of a nascent field. Build. Environ.
**2016**, 97, 196–202. [Google Scholar] [CrossRef] [Green Version] - Menapace, A.; Righetti, M.; Santopietro, S.; Gargano, R.; Dalvit, G. Stochastic characterisation of the district heating load pattern of residential buildings. Euroheat Power
**2019**, 16, 14–19. [Google Scholar] - Gargano, R.; Tricarico, C.; DEL Giudice, G.; Granata, F. A stochastic model for daily residential water demand. Water Supply
**2016**, 16, 1753–1767. [Google Scholar] [CrossRef] [Green Version] - Catalina, T.; Virgone, J.; Blanco, E. Development and validation of regression models to predict monthly heating demand for residential buildings. Energy Build.
**2008**, 40, 1825–1832. [Google Scholar] [CrossRef] - Wojdyga, K. An influence of weather conditions on heat demand in district heating systems. Energy Build.
**2008**, 40, 2009–2014. [Google Scholar] [CrossRef] - Pacheco-Torres, R.; Ordóñez, J.; Martínez, G. Energy efficient design of building: A review. Renew. Sustain. Energy Rev.
**2012**, 16, 3559–3573. [Google Scholar] [CrossRef] - Popescu, D.; Ungureanu, F.; Hernández-Guerrero, A. Simulation models for the analysis of space heat consumption of buildings. Energy
**2009**, 34, 1447–1453. [Google Scholar] [CrossRef] - Amjady, N. Short-term hourly load forecasting using time-series modeling with peak load estimation capability. IEEE Trans. Power Syst.
**2001**, 16, 798–805. [Google Scholar] [CrossRef] - Di Lascio, F.M.L.; Menapace, A.; Righetti, M.; Maurizio, R. Joint and conditional dependence modelling of peak district heating demand and outdoor temperature: A copula-based approach. J. Ital. Stat. Soc.
**2019**, 29, 373–395. [Google Scholar] [CrossRef] - Di Lascio, F.M.L.; Menapace, A.; Righetti, M. Analysing the relationship between district heating demand and weather conditions through conditional mixture copula. Environ. Ecol. Stat.
**2021**, 28, 53–72. [Google Scholar] [CrossRef] - Wu, L.; Kaiser, G.; Solomon, D.; Winter, R.; Boulanger, A.; Anderson, R. Improving efficiency and reliability of building systems using machine learning and automated online evaluation. In Proceedings of the 2012 IEEE Long Island Systems, Applications and Technology Conference, Farmingdale, NY, USA, 4 May 2012; pp. 1–6. [Google Scholar] [CrossRef] [Green Version]
- Kottegoda, N.T.; Rosso, R. Applied Statistics for Civil and Environmental Engineers; Blackwell: Malden, MA, USA, 2008. [Google Scholar]
- Gumbel, E.J. The Return Period of Flood Flows. Ann. Math. Stat.
**1941**, 12, 163–190. [Google Scholar] [CrossRef] - Pal, R.; Pani, P. Seasonality, barrage (Farakka) regulated hydrology and flood scenarios of the Ganga River: A study based on MNDWI and simple Gumbel model. Model. Earth Syst. Environ.
**2016**, 2, 1–11. [Google Scholar] [CrossRef] [Green Version] - Tricarico, C.; De Marinis, G.; Gargano, R.; Leopardi, A. Peak residential water demand. Proc. Inst. Civ. Eng. Water Manag.
**2007**, 160, 115–121. [Google Scholar] [CrossRef] [Green Version] - Gargano, R.; Tricarico, C.; Granata, F.; Santopietro, S.; De Marinis, G. Probabilistic Models for the Peak Residential Water Demand. Water
**2017**, 9, 417. [Google Scholar] [CrossRef] [Green Version] - DPR Decreto del Presidente della Repubblica n 74 (16 Aprile 2013) Regolamento Recante Definizione dei Criteri Generali in Materia Desercizio, Conduzione, Controllo, Manutenzione e Ispezione degli Impianti Termici per la Climatizzazione Invernale ed Estiva degli Edifici e per la Preparazione dell’acqua Calda per usi Igienici Sanitari, a Norma dell’articolo 4, Comma 1, Lettere a) e c), del d.lgs. 19 Agosto 2005, n.192 della Repubblica n. 74. Available online: http://www.energia.provincia.tn.it/binary/pat_agenzia_energia/dpr%2074-2013.pdf (accessed on 27 August 2021).
- Gadd, H.; Werner, S. Daily heat load variations in Swedish district heating systems. Appl. Energy
**2013**, 106, 47–55. [Google Scholar] [CrossRef] [Green Version] - Phillips, P.C.B.; Perron, P. Testing for a unit root in time series regression. Biometrika
**1988**, 75, 335–346. [Google Scholar] [CrossRef] - Massey, F.J., Jr. The Kolmogorov-Smirnov test for goodness of fit. J. Am. Stat. Assoc.
**1951**, 46, 68–78. [Google Scholar] [CrossRef] - Goia, A.; May, C.; Fusai, G. Functional clustering and linear regression for peak load forecasting. Int. J. Forecast.
**2010**, 26, 700–711. [Google Scholar] [CrossRef] - Seber, G.A.F.; Lee, A.J. Linear regression analysis; John Wiley & Sons: Hoboken, NJ, USA, 2012; Volume 329. [Google Scholar]

**Figure 1.**Map of the selected sample of residential users fed by the DH of Bozen-Bolzano in (

**a**), and map of the load-based user groups in (

**b**).

**Figure 2.**Map of the selected sample of residential users fed by the DH of Bozen-Bolzano in (

**a**), and map of the load-based user groups in (

**b**).

**Figure 3.**The daily standard deviation of the daily heat load pattern on the y-axis and the days of the heating season on the x-axis of user 4 belonging to group 2 and user 23 belonging to group 1.

**Figure 5.**$\mu $-$CV$ relationship of the daily heat load pattern (${w}_{d}$) with parameter P for the two considered groups ((

**a**) group 1, (

**b**) group 2); the dots show the real data values and their colour the corresponding parameter P (colormap), while the lines indicate the results of Equation (11) with the parameters P as indicated in the legend.

**Figure 6.**Comparison between the three cumulative distribution functions Log-Normal (LN), Log-Logistic (LL), and Gumbel (GU), and the observed cumulated frequencies of the daily heat load pattern ${w}_{d}$ (-) of three users at different times of the day.

**Figure 7.**Seasonal heat load pattern during the period of the heating season versus outside temperature (°C) and linear regression trend (red line); group 1 in (

**a**), and group 2 in (

**b**).

**Figure 8.**Graphs about residuals analysis: (

**a**) residuals versus fitted values, (

**b**) QQ-plot, (

**c**) standardised residuals versus fitted values, and (

**d**) residuals versus average.

**Figure 9.**District heat load forecasting of users 51 and 56 belonging to group 1 and 2, respectively. (

**a**,

**c**,

**e**) concerns user 51, while (

**b**,

**d**,

**f**) user 56. (

**a**,

**b**) show the plot of the seasonal component, (

**c**,

**d**) indicate the intra-daily patterns of the heating season with the mean and boundaries (5th and 95th percentiles, respectively), and (

**e**,

**f**) describe the final heat load prediction of three days.

a | b | c | |
---|---|---|---|

Group 1 | 1 | 0.1 | 0.1 |

Group 2 | 1.5 | 0.05 | 0.15 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Menapace, A.; Santopietro, S.; Gargano, R.; Righetti, M.
Stochastic Generation of District Heat Load. *Energies* **2021**, *14*, 5344.
https://doi.org/10.3390/en14175344

**AMA Style**

Menapace A, Santopietro S, Gargano R, Righetti M.
Stochastic Generation of District Heat Load. *Energies*. 2021; 14(17):5344.
https://doi.org/10.3390/en14175344

**Chicago/Turabian Style**

Menapace, Andrea, Simone Santopietro, Rudy Gargano, and Maurizio Righetti.
2021. "Stochastic Generation of District Heat Load" *Energies* 14, no. 17: 5344.
https://doi.org/10.3390/en14175344