Probabilistic Calibration of Building Energy Models : For Scalable and Detailed Energy Performance Assessment of District-Heated Multifamily Buildings

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ISBN 978-91-7485-473-2 Address: P.O. Box 883, SE-721 23 Västerås. Sweden

Probabilistic Calibration of Building

Energy Models

For Scalable and Detailed Energy Performance

Assessment of District-Heated Multifamily Buildings

Lukas Lundström

Mälardalen University Doctoral Dissertation 318

Lu ka s L u n d str ö m P R OB A B IL IS TIC C A LIB R A TION OF B U IL D IN G E N ER G Y M O D EL S 2020

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Mälardalen University Press Dissertations No. 318

PROBABILISTIC CALIBRATION OF BUILDING ENERGY MODELS

FOR SCALABLE AND DETAILED ENERGY PERFORMANCE ASSESSMENT OF DISTRICT-HEATED MULTIFAMILY BUILDINGS

Lukas Lundström 2020

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This thesis is based on work conducted within the industrial post-graduate school Reesbe –Resource-Efficient Energy Systems in the Built Environment. The projects in Reesbe are aimed at key issues in the interface between the business responsibilities of different actors in order to find common solutions for improving energy efficiency that are resource-efficient in terms of primary energy and low environmental impact.

The research groups that participate are Energy Systems at the University of Gävle, Energy and Environmental Technology at the Mälardalen University, and Energy and Environmental Technology at the Dalarn University. Reesbe is an effort in close co-operation with the industry in the three regions of Gävleborg, Dalarna, and Mälardalen, and is funded by the Knowledge Foundation (KK-stiftelsen).

www.hig.se/Reesbe

Copyright c Lukas Lundström, 2020

ISBN 978-91-7485-473-2 ISSN 1651-4238

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Mälardalen University Press Dissertations No. 318

PROBABILISTIC CALIBRATION OF BUILDING ENERGY MODELS

FOR SCALABLE AND DETAILED ENERGY PERFORMANCE ASSESSMENT OF DISTRICT-HEATED MULTIFAMILY BUILDINGS

Lukas Lundström

Akademisk avhandling

som för avläggande av teknologie doktorsexamen i energi- och miljöteknik vid Akademin för ekonomi, samhälle och teknik kommer att offentligen försvaras torsdagen den 10 september 2020, 10.00 i Milos + digital (Zoom), Mälardalens högskola, Västerås.

Fakultetsopponent: Angela Sasic, Chalmers University of Technology

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Abstract

There is a global need to reduce energy consumption and integrate a larger share of renewable energy production while meeting expectations for human well-being and economic growth. Buildings have a key role to play in this transition to more sustainable cities and communities.

Building energy modeling (BEM) and simulation are needed to gain detailed knowledge ofthe heat flows and parameters that determine the thermal energy performance of a building. Remote sensing techniques have enabled the generation of geometrical representations of existing buildings on the scale of entire cities. However, parameters describing the thermal properties ofthe building envelope and the technical systems are usually not readily accessible in a digitized form and need to be inferred. Further, buildings are complex systems with indoor environmental conditions that vary dynamically under the stochastic influence of weather and occupant behavior and the availability of metering data is often limited. Consequently, robust inference is needed to handle high and time-varying uncertainty and a varying degree of data availability.

This thesis starts with investigation of meteorological reanalyses, remote sensing and onsite metering data sources. Next, the developed dynamic and physics-based BEM, consisting of a thermal network and modeling procedures for the technical systems, passive heat gains and boundary conditions, is presented. Finally, the calibration framework is presented, including a method to transform a deterministic BEM into a fully probabilistic BEM, an iterated extended Kalman filtering algorithm and a probabilistic calibration procedure to infer uncertain parameters and incorporate prior knowledge. The results suggest that the proposed BEM is sufficiently detailed to provide actionable insights, while remaining identifiable given a sufficiently informative prior model. Such a prior model can be obtained based solely on knowledge of the underlying physical properties of the parameters, but also enables incorporation of more specific information about the building. The probabilistic calibration approach has the capability to combine evidence from both data and knowledge-based sources; this is necessary for robust inference given the often highly uncertain reality in which buildings operate.

The contributions of this thesis bring us a step closer to producing models of existing buildings, on the scale of whole cities, that can simulate reality sufficiently well to gain actionable insights on thermal energy performance, enable buildings to act as active components of the energy system and ultimately increase the operational resilience of the built environment.

ISBN 978-91-7485-473-2 ISSN 1651-4238

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The theory of probabilities is basically only common sense reduced to a calculus. — Pierre-Simon Laplace

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Sammanfattning

För att minska miljöpåverkan och den globala uppvärmningen behöver ener-gianvändningen effektiviseras och en högre andel förnybar och ofta variabel energiproduktion integreras, samtidigt som ekonomisk tillväxt och människors välbefinnande behöver tillgodoses. Energieffektivisering och digitalisering av byggnader spelar en viktig roll för att möjliggöra att av dessa mål nås.

Modellering och simulering av byggnaders energianvändning behövs för att få detaljerad kunskap om de värmeflöden och parametrar som avgör en byg-gnads energiprestanda. Framsteg inom flygburen laserskanning har möjlig-gjort skapandet av geometriska modeller av det befintliga byggnadsbeståndet. Parametrar som beskriver byggnaders termiska egenskaper och de tekniska systemen är emellertid ofta inte tillgängliga i ett digitaliserat format och be-höver estimeras. Byggnader är komplexa system där temperaturer och ener-gianvändning varierar dynamiskt under stokastiskt inflytande av väder, hur de brukas, och egenskaper i delsystem och komponenter; och trots framsteg i dig-italiseringen så är tillgång till sensordata från byggnader ofta begränsat. Föl-jaktligen behöver robusta estimeringsmetoder kunna hantera varierande grader av osäkerhet och datatillgänglighet.

Denna avhandling börjar med att undersöka tillgänglig information från meteorologiska reanalyser, fjärranalys och mätdata från byggnaden. Därefter presenteras en dynamisk och fysikbaserad byggnadsenergimodell. Slutligen presenteras en modellkalibreringsmetod bestående av en filtreringsalgoritm för att hantera osäkra tidsserier och en probabilistisk inferensmetod för att hantera osäkra parametrar och assimilera kunskapsbaserad information.

Resultaten visar att byggnadsenergimodellen är detaljrik nog för att ge an-vändbara kunskaper om de delsystem och komponenter som avgör byggnadens faktiska energiprestanda, men fortfarande identifierbar med hjälp information som kan erhållas enbart baserad på kunskap om modellparameternas fysiska egenskaper. Vidare har den förslagna kalibreringsmetoden förmågan att as-similera information från både data- och kunskapsbaserade källor; vilket är en förutsättning för robust och skalbar inferens givet den stokastiska och icke-digitaliserade verklighet de flesta byggnader verkar i.

Bidragen från denna avhandling tar ett steg närmare till att skapa modeller av existerande byggnader, för hela städer, vilka kan simulera byggnaderna till-räckligt verklighetstroget för att ge analyserbara kunskaper om byggnaders energiprestanda, möjliggöra byggnader till att vara aktiva komponenter i en-ergisystemet och på sikt bidra till att skapa förutsättningar för övergången till mer hållbara och smarta städer.

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List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Heat Demand Profiles of Energy Conservation Measures in

Buildings and Their Impact on a District Heating System. Published in Applied Energy (Lundström and Wallin, 2016).

II Mesoscale Climate Datasets for Building Modelling and

Simulation. Published in Proceedings of CLIMA 2016: 12th REHVA

World Congress(Lundström, 2016)

III Adaptive Weather Correction of Energy Consumption Data.

Published in Energy Procedia, Proceedings of ICAE 2016:

International Conference on Applied Energy. (Lundström, 2017)

IV Development of a Space Heating Model Suitable for Automated

Model Generation of Existing Multifamily Buildings — Case Study in Nordic Climate. Published in Energies (Lundström, Akander, and Zambrano, 2019)

V Uncertainty in Hourly Readings from District Heat Billing Meters.

Proceedings of SIMS 2019: 60th International Conference of

Scandinavian Simulation Society(Lundström and Dahlquist, 2020)

VI Bayesian Calibration with Augmented Stochastic State-Space

Models of District-Heated Multifamily Buildings. Published in

Energies(Lundström and Akander, 2019)

Paper I was previously included in the licentiate thesis entitled "Heat Demand Profiles of Buildings’ Energy Conservation Measures and Their Impact on Re-newable and Resource Efficient District Heating Systems" (Lundström, 2016)

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Nomenclature

Abbreviations

AHU Air Handling Unit

BEM Building Energy Modeling

DH District Heating

DCW Domestic Cold Water

DHW Domestic Hot Water

DHWC Domestic Hot Water Circulation

HMC Hamiltonian Monte Carlo

LiDAR Light Detection And Ranging

MLE Maximum Likelihood Estimation

RMSD Root Mean Square Deviation

TRV Thermostatic Radiator Valve

PDF Probability Density Function

Symbols

∆t time interval [h]

αsol solar absorption [-] or solar altitude [◦]

η efficiency [-]

γ azimuth angle [◦]

κ areal heat capacity [Wh/(◦C · m2s)]

κ ρa heat capacity of air per volume [Ws/(l ·◦C)]

φ normalized thermal power [W/m2_fl]

ρa density of air [kg/m3]

σ standard deviation

σ Stefan-Boltzmann constant, 5.67e-8 [W/(m2· K4_)]

θ centigrade temperature [◦C]

θ set of unknown/uncertain parameters

π probability density distribution

ε Kalman filter innovation

A continuous-time state transition matrix

B continuous-time input coefficient matrix

F state transition matrix

G input coefficient matrix

H measurement model matrix

K Kalman gain

P covariance matrix

Q process noise covariance

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R measurement noise covariance

S Kalman filter innovation covariance

u input vector

v measurement noise vector

w Process noise vector

x state vector

y measurement vector

C coefficient [-] or normalized heat capacity [Wh/(◦C · m2_fl)] D, H, L distance, height, length [m]

F, f factor/fraction [-]

H information entropy

H normalized heat transfer coefficient [W/(◦C · m2_fl)]

H∗ modified building’s ceiling height [m]

I solar or thermal irradiance [W/m2_s]

L quantization levels

P building perimeter [m]

Q quantity of thermal energy [kWh]

R normalized thermal resistance [m2_fl·◦C/W]

U thermal transmittance [W/(◦C · m2_s)]

Uloc, U10m local wind speed [m/s], meteorological wind speed at 10 m height [m/s]

g total solar energy transmittance [-]

h surface coefficient of heat transfer [W/(◦C · m2_s)]

n exponent

qV specific air flow rate [l/(s · m2fl)]

q_V∗ potential specific air flow rate [Pan_/m2 fl] r ratio [-] u control signal [-] Subscripts b building or base bl (window) blinds

c, ci convective, convective interior surface di f, dir diffuse, direct

e external (as in outdoor)

el (building) element

ew external walls

g f ground floor

gl glazing (windows, doors etc)

gr ground

hor horizontal

dhe domestic household electricity

hyd hydronic heating system

im internal mass (internal walls, intermediate floors and adiabatic external walls) in f infiltration (uncontrolled air leakage)

int internal (as in indoor)

k time index, discrete-time

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lmtd radiator logarithmic mean temperature difference

m mass related conductance or capacitance

max maximum

op operative

obst, ovh obstacle, overhang

p pipe

pb proportional band

r, ri radiative, radiative interior surface

ret return

r f roof

s stack

se, si surface exterior, surface interior

set set-point

sh shading or sheltering

sky sky temperature or sky thermal radiation

sol solar radiation/heat gain

strd surface thermal radiation downwards

sup supply

sys system

t time index, continuous-time

tb thermal bridges

tot total

trv thermostatic radiator valve(s)

ve ventilation

ver vertical

vi virtual ground layer

w wind

ww windward-oriented

wi windows

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Acknowledgments

This research has been carried out under the auspices of the industrial post-graduate school Reesbe where the three participating universities (University of Mälardalen, University of Gävle, University of Dalarna), together with the Knowledge Foundation (KK-stiftelsen), the participating companies (Eskil-stuna Kommunfastigheter and Eskil(Eskil-stuna Energy and Environment) and Es-kilstuna City all contributed to make this dissertation possible.

I would like to thank my supervisors Erik Dahlquist, Fredrik Wallin and Jan Akander for their guidance and support. Extra thanks go to Jan Akander for your co-authorship in my later publications, where I felt we were able to really push that boundary and contribute with new knowledge. I would like to thank Jesús Zambrano for insightful discussions on mathematical modeling and Björn Karlsson for interesting discussions and recommending me for this PhD candidate position.

I would like to thank Eskilstuna Kommunfastigheter for the opportunity to conduct research while gaining work experience as both project manager and energy strategist. Jan Helgesson, as company mentor, has always made me feel welcome and ensured I was in positions where I could develop professionally and gain insights of challenges relevant for the research. All my colleagues at Mälardalens University, Eskilstuna Kommunfastigheter and Reesbe research school deserve a huge thank you for fruitful meetings, fikas, after work beers and support with both work- and research-related matters.

My greatest and most important supporters, however, are those I have at home. Jessika and Lovis, thank you for all the joy and love you have given me.

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1. Introduction

This chapter introduces the background and motivation behind the work in this thesis and discusses related previous research. Further, the identified research gaps, research questions, objective and delimitation, and outline of this thesis are presented.

1.1 Background

In the past century, continuously increasing access to abundant, cheap and high-density energy sources has been a prerequisite and a driving force for the growth of modern society. In 2019, global energy use was ten times higher than in 1919, and the share from fossil fuels was about 80% [8]. The dom-inant share of fossil fuel based energy is a major driver of global warming, which is the result of an increase of the greenhouse effect due to increased concentrations of CO2 and other greenhouse gases in the atmosphere. Human well-being, technological development and economic growth have thus be-come dependent on a non-sustainable harvesting of limited resources. There is a need to reduce energy consumption, integrate a larger share of renewable en-ergy production and mitigate environmental impacts, all while meeting expec-tations for human well-being and economic growth. In the European Union, the buildings sector accounts for 40% of final energy use [9]; energy-efficient buildings have a key role to play in this transition to a more sustainable society.

1.1.1 District heating systems

In Sweden, most cities have extensive district heating (DH) networks. DH operators have a share of about 55% of the heating market in the buildings and service sector, and a share of 90% when considering only multifamily buildings [10].

District heating is a technical system with centralized heat production and a network of pipelines using hot water as an energy carrier to distribute heat to end users. DH enables the utilization of lower-value resources such as in-dustrial surplus heat, heat from bulky residual biomass and waste materials, thereby contributing to improved total energy system efficiency by avoiding the use of high-exergy energy. Resource utilization can be further improved by co-generation of heat and electricity in combined heat and power (CHP) plants. Figure 1.1 shows the merit order of operation of the plants and the heat load duration for the Eskilstuna DH system.

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0 50 100 150 200 250 37 170 365 MW Time, days Oil-HO Turbine bypass Bio-HO Bio-CHP Heat load Electricity prod.

Peak Mid Base

Figure 1.1. Duration plot showing the heat and electricity production (average daily values) of the district heating system of Eskilstuna, Sweden. The marginal load is divided into three periods: base, mid and peak load.

Peak Mid Base Peak Mid Base a) duration plot b) temperature correlation plot

37 170 360 0.0 0.1 0.2 0.3 -15.0 -3.0 5.5 22.0

Time, days Outdoor temperature, °C

Avg. daily thermal power, kW

Building

envelope Householdelectricity Domestichot water

Figure 1.2. Heat demand profiles for three energy-saving measures for a typical Swedish multifamily building. The DH marginal production type (base, mid and peak) is indicated with shading in the background. The profiles are scaled to the annual total of 1 MWh.

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Knowledge on how energy retrofitting of district-heated buildings impacts the DH system, which was the objective of the Licentiate thesis [7] and Paper I, is essential for planning of sustainable communities. Figure 1.2 shows the impact on the Eskilstuna DH system of a selection of energy-saving measures. These studies concluded that energy-conservation measures that decrease heat loss due to heat transfer (for example, by improving insulation of the build-ing envelope and heat recovery in the ventilation) lead to the most seasonally evenly distributed heat load curve for the DH system. A more even heat load across the seasons increases the utilization rate of the base plant, improves the conditions for electricity co-generation and makes more on-demand capacity available that could potentially be used to integrate the growing share of in-termittent and weather-dependent electricity production in the Nordic energy system. Thus, although buildings utilize large quantities of energy, there are possibilities to reduce their energy use both with energy efficiency measures that are integrated when they are refurbished, and by supplying energy in a more efficient way.

1.1.2 Actual energy performance of buildings

Buildings are complex systems with indoor environmental conditions and en-ergy use that vary dynamically under the stochastic influences of weather, oc-cupant behavior and component and equipment failures [11]. Consequently, there is often a discrepancy between the intended and actual energy perfor-mance of buildings, commonly known as the ’energy gap’ [12]. Senave et al. [13] defined three key elements for which thorough insight is required to asses energy performance of existing buildings: (i) the thermal performance of the building fabric, (ii) the efficiency of the technical systems, and (iii) the behav-ior of the users.

Solar heat gains Internal gains Heating system Internal DHWC Space heating Overheating Thermal mass Ventilation Transmission

Figure 1.3.Heat balance of a district-heated Swedish multifamily building. The height of the gray areas marks the relative share of the total heat flow over the heating season. Figure 1.3 shows the heat balance of a typical district-heated Swedish mul-tifamily building (from the case study of Paper VI). The middle bar of the figure represents the conditioned space, the left bars show the passive heat

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District heating

Solar heat gains Internal gains Heating system DHW Internal DHWC Space heating Ventilation Transmission Inﬁltration Manual venting AHU Thermal bridges Roof Windows External walls Ground ﬂoor

Pipe network _{Internal mass}

Figure 1.4. Typical heat flows (over the heating season) in a DH-connected Swedish multifamily building.

gains and the controlled heat gain from the heating system, and the right bar shows the heat losses. The gains and losses need to be in balance to maintain the desired temperature of the conditioned space. ’Overheating’ in the figure represents the heat from the heating system that causes indoor temperatures to rise above the desired set-point temperature, heat use that could potentially be avoided with improved control. The ’thermal mass’ represents the heat flow going back and forth between the conditioned space and the thermal mass of building. Identifying the characteristics of the thermal mass is needed to achieve good control of thermal comfort and for application of more flexible heat use. According to Kensby et al. [14], there is untapped potential in us-ing thermal mass as short-term thermal energy storage to shift heat demand from times when the district heat is produced at high cost and with negative environmental impact to more favorable hours.

In Figure 1.4, the heat balance of Figure 1.3 is expanded on the left side to also show the total heat supply, segregated into heat used for the heating system, domestic hot water (DHW), domestic hot water circulation (DHWC) and losses in the behind-the-meter piping network. Metering is often avail-able only for the total supplied heating, but more the detailed segregation is required to normalize energy use when for example reporting performance ratings for code compliance and green certification. Further, these heat flows have distinct characteristics in terms of required temperature levels, weather dependency and time patterns; information that is of interest for the DH opera-tor. On the right side of Figure 1.4, the heat losses are further segregated at the level of subsystems and components. This is typically the level of detail that building owners are interested in when identifying and applying energy-saving measures.

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1.1.3 Building energy modeling

The total heating use is often the only heat flow that is metered and thus build-ing energy modelbuild-ing (BEM) is required to identify other heat flows in the building and the parameters governing these flows.

BEM is often categorized into two major approaches: law-driven (forward) and data-driven (inverse) [15]. Law-driven models apply a given set of phys-ical laws that describe the system performance, whereas data-driven mod-els use system behavior as a predictor for system performance. Examples of building engineering-based energy simulation software that utilize a

law-driven approach include EnergyPlus [16] and IDA ICE1. Engineering-based

tools can offer flexibility regarding modeling the building’s subsystems and components. Disadvantages of engineering-based tools include their require-ment for detailed information input and model configuration and that they tend to model how the building is designed to operate and not how it actually op-erates. Data-driven modeling approaches have gained considerable attention in recent years; the review by Amasyali and El-Gohary [17] identified more than 50 articles using data-driven approaches. In contrast to law-driven model-ing, data-driven models must be trained or calibrated, making use of metering data to determine model parameters and thereby resulting in robust simulated predictions of energy use. Data-driven approaches have the advantage of re-quiring less information input about the building itself. However, data-driven approaches may not perform well outside their training range and they often yield limited knowledge of underlying aspects that govern the energy use.

Gray-box modeling combines law-driven and data-driven approaches, thereby leveraging the advantages and minimizing the disadvantages of both approaches. Gray-box models have parameters that are based on building physics and are therefore interpretable as entities that describe the characteristics of the build-ing’s energy performance. However, they are often simplified and lumped to an extent that they cannot reveal many details about the building’s subsystems and components. Gray-box models are calibrated to better match a building’s actual operation performance and can potentially also provide reasonable re-sults outside the range of the training data.

Figure 1.5 shows a conceptualization of the relationship between model complexity of above discussed model types, the required amount of informa-tion input and the potential insights the model can provide regarding the actual energy performance of the building.

1.1.4 Calibration

In a review on methodologies and recent advancements in the calibration of BEMs, Fabrizio and Monetti [18] concluded that automated models are of-ten simplified in order to reduce computational time, and as a result, more

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Increasing complexity Required information input Potential insights <-Less Mor e ->

Figure 1.5.Conceptualization of the relationship between model complexity, required amount of information input and the potential insights the model can provide on the actual energy performance of the building.

complex models are difficult to handle in the calibration process. They also highlighted the importance of assessment of occupant behavior, “since build-ing usage is one of the main sources of uncertainty in buildbuild-ing simulation models” [18].

In recent years, stochastic calibration methods have become the standard approach; they do not rely on single deterministic values for uncertain quanti-ties, but instead incorporate and explore a distribution of possible values [19]. Using the Bayesian approach allows incorporation of any prior information or knowledge that could help in the inference of unknown parameters; informa-tion which is often available from literature, standards, energy performance audits or the modeler’s knowledge of plausible values based the underlying physics or previous experience. According to Chong and Menberg [20], the information content within datasets and the quality of prior knowledge limits the number of uncertain parameters that can be introduced in the model.

Previous BEM calibration research can be grouped into research that uses relatively simple state-space models with 2–4-state variables (for example Bacher and Madsen [21], Coffman and Barooah [22], Raillon and Ghiaus [23], Rouch-ier et al. [24]) and those working with calibration of detailed engineering-based models (for example, Chong and Menberg [20], Tian et al. [25], Ban-dera and Ruiz [26]). The benefits of simpler state-space models are that little prior information about the building is needed and that they can be efficiently computed and used directly in Monte Carlo Markov chain-based inference. For engineering-based tools, the typical Bayesian approach is to approximate the detailed model with a simpler surrogate or emulator model [20].

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1.1.5 Time-varying uncertainty

A distinction is often made between two types of uncertainty: aleatory and epistemic uncertainty [27]. Aleatory uncertainly is due to inherent or natural variation of the system, and is often also called variability, or stochastic or irreducible uncertainty. Epistemic uncertainty arises from lack of knowledge, and is also called state of knowledge, subjective or reducible uncertainty [27] or, more expressively, as ‘things we could in principle know but don’t in prac-tice’ [28]. In the context of BEM, occupant behavior-induced uncertainties are large sources of epistemic and time-varying uncertainty. The modeling might be improved, and more metering data could be gathered, but in practice it is impossible to fully reduce occupant behavior-induced uncertainty.

Solar heat gains, internal heat gains, and heat use for domestic hot water are inputs that contain high levels of time-varying uncertainty. Internal heat gains and DHW use are uncertain due to their dependency on occupant be-havior [28], whereas solar heat gains are uncertain due to both uncertainty of occupant behavior (such as shading operation) and the complexity involved in modeling solar irradiance and shading from the surroundings [IV]. Aug-mented stochastic state-space models have been suggested as a suitable ap-proach to deal with such highly non-deterministic inputs: Kim and Park [29] used a detailed augmented state-space model (consisting of 15 states) to es-timate time-varying process disturbances attributed to uncertainty in internal heat sources and airflow, and Coffman and Barooah [22] augmented a simpli-fied two state-space model with a third disturbance state to account for unmea-sured disturbances attributed to occupant behavior.

1.2 Identified research gaps

A gap in the current state-of-the-art was identified in the lack of a BEM that can simulate reality sufficiently well to provide actionable insights while still allowing for direct calibration in a scalable way. Figure 1.6 shows the iden-tified gap that the proposed BEM aims to fill. This BEM would be closer to established engineering-based simulation tools with respect to complexity and potential to provide insights about the actual energy performance of the build-ing on the level of detail visualized in Figure 1.4, but would still be able to be directly calibrated (without an intermediate emulator step) in a similar manner as the 2–3-state gray-box BEM that is common in the literature. The method was first developed in Paper IV, further enhanced in Paper VI and is presented and discussed in Chapter 3.

Classic stochastic state-space modeling accounts for uncertainty in the out-puts and in the states, while inout-puts are considered as deterministic. Previous research has shown that state augmentation can be used to account for unmea-sured disturbances. However, further research was needed into how to achieve a fully probabilistic BEM in which inputs are also defined with time-varying

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Increasing complexity Required information input Potential insights Identiﬁable / directly calibratable <-Less Mor e ->

Figure 1.6.Conceptualization of the relationship between model complexity, required amount of information input, identifiability and potential insights.

uncertainty. Further investigation was also needed into how to obtain and model the time-varying uncertainty of the inputs. The method was developed in Paper VI and is included in Chapter 4.

Bayesian inference has gained attention in the BEM calibration literature because of its ability to handle and propagate uncertainty, but it has so far been used only with simpler models or with an intermediate emulator step when used with engineering-based tools. In Paper VI a framework for performing such direct Bayesian inference (without an intermediate emulator step) was developed; it is presented and discussed in Chapter 4.

Large-scale construction and calibration of detailed BEMs requires system-atically available and informative data sources. Practical knowledge (methods and tools) on how to utilize meteorological reanalysis and satellite-based solar irradiance data sources for BEM simulations were missing at the onset of this thesis work. Use of such data source has been a recurring topic in all the ap-pended papers, Chapter 2 summaries the main findings. Knowledge regarding the on-site metering data that is available for the targeted building stock was also identified as a gap. Paper V investigated the quality of hourly readings from DH-billing meters (included in Chapter 2). Paper VI investigated the usability of hourly indoor air temperature sensor readings and hourly billing meter readings for domestic household electricity, domestic cold-water and domestic hot-water billing (included in Chapter 2 and Chapter 4).

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1.3 Research questions

Based on the identified research gaps, the following research questions were formed:

(Q1) What information from meteorological reanalysis, remote sensing and on-site metering data sources can be utilized for BEM calibra-tion in a scalable way?

(Q2) What level of detail is required to obtain a BEM that can provide actionable insights on the building’s actual energy performance, yet still allow calibration in a scalable way?

(Q3) What are the advantages and disadvantages of using a probabilis-tic BEM calibration approach?

1.4 Objective and delimitation

The objective of this thesis is to provide a framework for probabilistic calibra-tion of BEMs that aims at fulfilling the following requirements:

• Result in actionable insights regarding aspects governing the actual energy performance of a building (including its subsystems and com-ponents)

• Direct calibration (without an intermediate emulator step) • Scalable to entire building stocks

• Integration of evidence from both data and knowledge-based sources, on both parameter and time-series level

• Handle uncertainties and situations of low data availability

The proposed framework was delimited to Swedish district-heated multi-family buildings. The targeted building stock is a suitable testing ground for large-scale BEM calibration, as it constitutes a large share of the Swedish building stock, is relatively homogeneous, and provides relatively good metering-data availability due to the DH billing meter infrastructure. With additional work and adoption, the proposed framework could also be applied to other building types.

1.5 Thesis outline

In addition to addressing the research questions, this thesis is written to func-tion as a reference for the developed BEM (Chapter 3) and the probabilistic calibration framework (Chapter 4). The contents of the chapters are as fol-lows.

Chapter 1 introduces the background of energy performance in district-heated multifamily buildings and the previous research and challenges faced

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in calibrating building energy models. Furthermore, research gaps, research questions, objectives and delimitation are presented.

Chapter 2 investigates the quality of hourly readings from DH billing me-ters (Paper V), describes the use of meteorological (re)analyses and satellite-derived solar irradiance as weather data sources in the BEM simulations (pa-pers II–VI) and investigates metering data sources carrying information about domestic hot water use (Paper VI).

Chapter 3 compiles, drawing from Paper IV and Paper VI, the determin-istic parts of the dynamic and physics-based BEM: the proposed thermal net-work; modeling of the technical systems of district-heated multifamily build-ing, solar and internal heat gains, and boundary conditions; and a case study comparing results of the BEM with the established simulation software IDA ICE.

Chapter 4 transforms the deterministic model of Chapter 3 into a proba-bilistic model, including time-varying uncertainty in inputs. Furthermore, this chapter presents: an iterated extended Kalman filtering algorithm for nonlinear state estimation; a probabilistic calibration procedure to assess parameter un-certainty and incorporate prior knowledge; a case study in which the proposed framework is tested with real buildings, the parameters are estimated with both Hamiltonian Monte Carlo sampling and penalized maximum-likelihood estimation optimization and the effect of different sources of time-series cali-bration data are investigated. This chapter is based on Paper VI.

Chapter 5 presents the main conclusions in relation to the research ques-tions and further discusses some aspects. Finally, the challenges and potential future work for the developed framework are addressed.

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2. Time-series data

This chapter investigates the quality of hourly readings from district heating billing meters (Paper V), describes the use of meteorological (re)analyses and satellite-derived solar irradiance as weather data sources in the BEM simu-lations (papers II–VI), and investigates metering data sources carrying infor-mation about domestic hot water use (Paper VI).

2.1 District heating

Many district heating (DH) operators collect hourly values from their heat billing meters in centralized databases [30,31]. Such data are of high value for for building energy model calibration. In Sweden, DH operators are re-quired to share daily meter readings with their customers [32], while hourly values need to be provided only if they are used in the pricing. Readings typically available in district heat meter data management systems are hourly averages of energy and flow and hourly instantaneous samples of the primary side supply and return temperatures [30]. Because of bandwidth constraints, only a finite number of bits are available, and recorded values are therefore quantized before transmission. Quantization can cause increased uncertainty, especially when the recorded value needs to be able to represent a large value range (for example, cumulative values). Therefore, hourly values transmitted from the heat meters can have quantization errors that are much larger than the accuracy of the measurement equipment. Such large quantization errors can severely deteriorate the quality of subsequent analyses.

2.1.1 Information entropy

Sandin et al. [30] suggested using information entropy ranking as a way of identifying heat meter readings with large quantization errors. Information entropy (H) is defined as the sum of the negative binary logarithm log₂(·) of the probabilities p(·) for each value xk in the time series of length n:

H= −

n

∑

k=1

p (xk) · log2(p (xk)) (2.1)

Two to the power of the entropy indicates the number of quantization lev-els (L) available in the meter readings: L ∝ 2H (provided that the observation period holds rich enough operation conditions). For example, H = 8 would indicate 256 levels and H = 4 would indicate 16 levels. However, information

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entropy depends on the actual operation conditions such as weather and the probability of an observation to occur (i.e. meters repeating same observation quantities, due to design or even weather conditions, will get a lower informa-tion entropy value). Thus, the 2Hestimate will generally result in fewer levels than what is available due to the meter configuration.

2.1.2 Uncertainty in hourly heat meters readings

The energy reading at time index k is denoted as Q1;k. The time-varying

un-certainty of the energy readings is estimated as σ1;k= σmet;k2 +

∆Q2

12 (2.2)

where σmet;kis the standard deviation (SD) at time index k due to uncertainty of the meter equipment and ∆Q denotes the quantization step size of the en-ergy readings (i.e. the largest unit of measure, typically 1, 10 or 100 kWh).

The ∆Q2/12 is a commonly used approximation of the variance for the

quanti-zation effect used for noise modeling [33]. The uncertainty calculations of the meter equipment is adopted from the European Standard CEN 1434-1:2015 [34]:

σmet;k=

q

σ2_f_;k + σ_t;k2 (2.3)

where σf;k is the standard deviations of the flow meter at time index k and

σt;kis the standard deviation of the temperature sensor pair and the calculator (where the temperature sensor pair is the dominating error source). Typical accuracy of Multical heat meters equipped with Ultraflow flow sensors (Kam-strup A/S, 2018) is used

σf;k= Q_1;k(1 + 0.01qV;p/qV;k) /100, if qV;k> qV;i ∆θ1;kqV;ic(1 + 0.011qV;p/qV;i) /100, if qV;k≤ qV;i σt;k= Q1;k(0.6 + 6/∆θ1;k) /100, if ∆θ1;k> 2 2qV;kc(0.6 + 6/2)/100, if ∆θ1;k≤ 2 (2.4)

where Q1;kis the energy reading at time step k (accumulated heat use between k−1 to k), qV;kis the flow rate reading at time step k (flow rate between k − 1 to k), qV;pis the permanent nominal flow rate , qV;iis the inferior flow rate (where the meter shall function without exceeding the allowed accuracy), c is the heat capacity of the fluids (assumed as a constant of 4.12 MJ/(K · m3) [35]), and ∆θ1;kis the average temperature difference between fluids calculated as

∆θ1;k= Q1;k/ (qV;k· c) (2.5)

The quantization error of flow readings is generally lower than for the en-ergy readings, especially during operation conditions when the temperature

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difference is low (see visualization in Figure 3). Therefore, in case of large quantization errors, it can be more accurate to estimated energy use from flow and temperature readings:

Q2;k= max (Q1;k− ∆Q, min (Q1;k+ ∆Q, c · ∆θ2;k· qV;k)) (2.6)

where ∆θ2;k denotes the temperature difference estimated from the

tempera-ture readings. Due to the instantaneous natempera-ture of the temperatempera-ture readings,

the ∆θ2;kapproximation can deviate much from the true average temperature

difference. While, eq. 2.5 can be assumed to calculate the true average ∆θ when the quantization error is negligible.

Following equation estimates the time-varying standard deviation for the energy use Q2: σ2;k= q (σ2;t;k)2+ σ2;qe;k 2 + (σmet;k)2 (2.7)

where σ2;t;kdenotes the uncertainty due to instantaneous nature of the temper-ature readings and σ2;qe;kdenotes the quantization error due to low resolution in the flow readings:

σ2;t;k= Qp/200 + 0.02 · Q2;k, σ2;qe;k=

∆θ2:k· ∆qV· c √

12 (2.8)

where Qp denotes the energy at nominal flow and ∆qV is the width of the

quantization step size of the flow readings.

The two energy variables Q1and Q2are not independent as they are derived from the same metering equipment. Therefore, the two quantities are weighted as two dependent normal variables [36]

Q₀= σ 2 2− ρσ1σ2 Q1+ σ12− ρσ1σ2 Q2 σ₁2+ σ₂2− 2ρσ1σ2 σ₀2= 1 − ρ 2 σ₁2σ₂2 σ₁2+ σ₂2− 2ρσ1σ2 (2.9)

where ρ denotes the correlation, which is assumed as

ρ = 0.5 · min (σ1, σ2) / max (σ1, σ2) (2.10)

The practical impact of assuming a dependency between the variables is a more conservative conflated estimate (both on mean and uncertainty interval) than if the variables would be assumed fully independent.

2.1.3 Case study

For paper V a larger dataset was gathered: consisting of hourly time series, from 266 DH billing meters serving multifamily buildings located in Eskil-stuna, Sweden, for the year 2018. Most of the heat meters are Kamstrup Mul-tical 601 / 602 calculators equipped with Kamstrup Ultraflow 54 / 34 ultra sonic flow meters.

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Figure 2.1.Information entropy ranking of 266 heat meters.

Figure 2.2. Three representative heat meters visualizing the impact of typically oc-curring quantization errors. Entropy of the hourly energy readings are given in the subtitles. Top row: energy vs temperature difference between supply and return flows. Middle and bottom rows: energy vs standard deviation (absolute and relative).

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Figure 2.1 shows the 266 heat meters ranked by their calculated entropy for both energy readings and flow readings. Figure 2.2 shows hourly energy readings and the time-varying uncertainty for three example DH meters. As can be seen in the figure, the uncertainty of the calculated energy quantity (blue dots) dominates under most conditions due to the high uncertainty in that the instantaneous temperature readings represent the average temperature difference for the whole integration step. However, when the quantization error is large, as for heat meter (a), the calculated energy quantity is a better estimate than the energy readings (black dots), especially during low energy use conditions. The conflated variable (purple dots) is a weighted estimate that weights the two variables according to their empirically estimated time-varying uncertainties.

2.2 Domestic hot water

Billing meter based DH readings typically consist of space heating, domes-tic hot water (DHW) use and heat losses. DHW use is inherently uncertain due to dependency on occupant behavior [28]. The availability and quality on metered hourly domestic hot water (DHW) varies greatly from building to building. In Paper VI three commonly occurring situations of data avail-ability regarding DHW were identified: (i) no hourly data, the DHW needs to be modeled; (ii) useful hourly domestic cold water metering exists; or (iii) actual hourly DHW use metering exists (metered for the whole building or aggregated from individual, per apartment, metering).

Recent Swedish building regulations require that new buildings and build-ings that undergo a major renovation have separate DHW metering. However, in practice, hourly metered DHW use is still often not available. In Sweden, domestic cold water (DCW) use is often gathered with the same infrastructure that is used for gathering DH use and such data are therefore available in a systematic way. However, the resolution of the hourly DCW values can be of poor quality (much of the current billing metering infrastructure was built in a time when only monthly values were required). Also, the placement of the DCW meters can be quite different than the placement of DH meters, as exemplified in Figure 2.3.

For paper VI individual DHW billing data was gathered. The dataset con-sists of hourly time series, grouped per apartment (799 in total), from 38 mul-tifamily buildings located in Eskilstuna, Sweden and was gathered from 2016 to 2018. Figure 2.4 shows probability density distributions for hourly DHW use based on that data. From the figure, it is apparent that there are diurnal and workday-related patterns. When averaging over many apartments, the proba-bility density tends to follow a close-to-normal distribution. However, when averaging over a smaller number of apartments, the long right-tailed distribu-tion is better described as log-normal.

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DCW meter 3 DCW meter 2 4 5 6 7 1 2 3 DH meter 1 DH meter 2 District heating Local heat Cold water Local cold water

DCW meter 1

Figure 2.3. District heating (DH) and domestic cold water (DCW) metering configu-ration for the case study buildings used in Paper VI.

N = 4 N = 20 N = 50 N = 799 0 40 80 120 0 10 20 30 40 0 10 20 30 0 5 10 15 23 22 21 20 19 18 17 16 15 14 13 12 11 109 8 7 6 5 4 3 2 1 0

Average, per apartment (N), hourly domestic hot water usage [l/h]

Hour of the da

y

Mon−Fri Sat−Sun

Figure 2.4. One year of DHW usage probability density distributions, grouped per hour and weekday: N denotes the number of apartments over which the measurements are averaged.

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2.3 Weather data

BEM calibration requires weather data describing the boundary conditions of the building site. The traditional approach has been to use data from nearby meteorological stations or on-site metering. This section describes the alter-native data sources meteorological (re)-analyses and satellite-derived solar ir-radiance.

2.3.1 Meteorological analyses and reanalyses

Meteorological weather forecasting centers use sophisticated methods for data assimilation, in which every few hours a previous forecast is combined with newly available observations in an optimal way to produce a new best esti-mate of the state of the atmosphere, called analysis, from which an updated, improved forecast is issued. The European Centre for Medium-Range Weather Forecasts’ (ECMWF) produces such analyses on a global scale (see Figure 2.5 for a schematic view). Reanalysis works in the same way, but aims to assimi-late historical observational data and spans a long time period that can extend back several decades or more.

Data assimilation Observation system

Observations Observations Observations

Forecast Forecast Forecast

Analysis Analysis Time Analysis Horizontal grid (latitude-longitude) Vertical grid (height or pressure) Medium-range forecast Forecast model

Physical process in model

Figure 2.5. The principle of the data assimilation process used for meteorological reanalyses.

The Swedish Meteorological and Hydrological Institute (SMHI) releases data from its operational mesoscale analysis product MESAN [37], which

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cov-ers the Scandinavian countries with a current horizontal resolution of 2.5 km (11 km and 22 km for data released before 2017). Based on cloud information produced by MESAN, SMHI also produces and releases hourly data for solar radiation quantities, a modeling system called STRÅNG [38]. MESAN and STRÅNG data were used in papers I, II and III. ERA5 is the fifth generation of the ECMWF global reanalyses and was released in 2016 [39]. This was the first of the ECMWF reanalyses to be produced as an operational service rather than a research project. ERA5 data are available from 1950 to near to the present time, at a global horizontal resolution of 31 km and hourly time steps [40]. ERA5 data was used in papers IV and VI. The variables air temper-ature, ground tempertemper-ature, wind speed, ground albedo and downward surface thermal radiation (used to derive sky temperature) were used.

Continuous development of meteorological (re)analysis systems results in higher spatial and temporal resolution and improved accuracy. Using these data sources for energy modeling enables consistent time series for locations where metered datasets are lacking or of poor quality. It is also possible to combine modeled and on-site metered data; for example, using metered data for temperature, which is cheap and simple to meter, and modeled data for solar, which is more expensive and cumbersome to meter.

2.3.2 Satellite-derived solar irradiance

When available, satellite-derived solar radiation sources perform better than those based on weather forecasts [41,42]. Solar irradiance has been estimated from satellite images since the beginning of the satellite era in the 1980s. Ei-ther geostationary or polar-orbiting satellites can be used. Geostationary satel-lites orbit in the equatorial plane synchronous with the rotation of the earth (and are thus stationary relative to the earth) and are placed high enough (35 000 km) to gather a full-disk view. Images from geostationary satellites have limited spatial resolution but high and continuous temporal resolution. Po-lar orbiting satellites are placed at a much lower altitude (800 km) and thus have a much higher spatial resolution. As they move relative to the earth, they cover the whole globe, but have a non-continuous temporal resolution (passing through a specific location twice a day). The principle for obtaining surface solar irradiance from satellite images is shown in subfigure (a) in Figure 2.6. In most cases, a cloud (pixel 2) appears brighter in the field of view of the satellite sensor than the ground (pixel 1). The main challenge lies in obtaining good estimates for what should be observed by the sensor for any pixel if the sky was clear [43].

Subfigure (b) in Figure 2.6 shows the coverage from the current Meteosat geostationary satellite placed at prime meridian and 36 000 km above the

equa-tor; the horizontal resolution is 0.05◦ (4–6 km for Europe) and the temporal

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Ground Space Pixel 1 Pixel 2 a) b) -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60

Direct normal radiation, SARAH-2, 2015-07-01 12:00

Figure 2.6. (a) The principle for obtaining satellite-based solar irradiance data. (b) Example irradiance image of the full-disk view from the SARAH dataset.

satellite viewing angle causes quality degradation [43]. The viewing angle shifts detected clouds northwards (southward in the southern hemisphere) and the clouds are seen from the side (the plane-parallel approximation becomes less accurate). The quality is especially reduced at low solar altitudes, in early morning and late afternoon, or in winter. Increased occurrence of snow also makes estimates less reliable due to the increased impact from surface snow/cloud differentiation schemes.

CAMS-rad [43] and SARAH [44] are two open data sources that are based on the Meteosat prime geostationary satellite images. CAMS-rad is more end-user friendly, as it can be accessed as time series readily via an API

inter-face1, from 2004 up to 2 days before the present time, with 15 minute

tem-poral resolution and any missing periods already handled. A peer-reviewed R client interface has been produced and released as open source by the author2. SARAH is released in batches, and is currently available for 1983 to 2017 with a 30 minute temporal resolution. CAMS-rad was utilized in papers IV and VI, mainly due to the necessity for recent data.

2.4 Discussion

Some of the benefits of using meteorological reanalysis such as ERA5 as a source for weather data are that the data are homogeneous, harmonized, have wide geographical coverage (global in the case of ERA5), provide vari-ables that are seldom measured locally (for example, solar irradiance, wind or sky temperature), and are often readily and publicly accessible. Satellite-derived solar radiation sources tend to perform better than those based on weather forecast engines. However, for regions not covered by the

geostation-1_{http://www.soda-pro.com/web-services/radiation/cams-radiation-service}

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ary placed satellites (such as northern Sweden), methods based on weather forecasts (such as ERA5 and STRÅNG) are required to obtain continuous time series. Further, if the intention is energy use forecasting, then the model calibration is likely to benefit from using reanalysis data based on the same weather forecast engine that is used for the forecast.

The information entropy ranking method, suggested by Sandin et al. [30], is suitable for identifying meters with large quantization errors. The method is straightforward to conduct as the only required inputs are the readings them-selves. However, it is dependent on the actual operation conditions, which makes it less suitable for comparison of meter readings from different time periods or different district heat operators.

District heat operators should aim for information entropy at a minimum of 5 bits per sample (approximately 32 observable quantization levels in a typi-cal year) to ensure qualitative hourly readings. To ensure high quality readings for the whole metering range and enable sub-hourly sampling, information en-tropy with at least 7 bits per hourly sample is required. For energy readings with entropy less than approximately 5 bits per sample, the suggested confla-tion method can counteract part of the quantizaconfla-tion error by merging infor-mation from the flow and the instantaneous temperature readings, especially during low energy use conditions.

The next generation of heat meters and data acquisition infrastructure [45] can provide data with higher resolution. However, it will take many years be-fore all current infrastructure is upgraded. Therebe-fore, the suggested conflation method can play a role in improving hourly readings for many years to come. The proposed conflation method assumes normal distributions. However, the quantization error is uniform and can also be biased [33]. The used additive

noise approximation (∆2/12) is only valid if ∆ is small compared with the

quantization levels L. The empirical uncertainty models are, however, likely to be larger error sources than the conflation method or the additive noise

approximation. Nevertheless, the proposed conflated energy quantity Q0 is

expected to be closer to the true mean values and have a tighter distribution

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3. A Dynamic Physics-Based Building Energy

Model

The model was first introduced in Paper IV and then further developed in Pa-per VI. In this chapter all the deterministic parts are extracted and presented in a coherent way: the proposed thermal network; modeling of the technical systems of district-heated multifamily buildings, solar and internal heat gains and boundary conditions. This chapter also presents a case study comparing results from the established simulation software IDA ICE.

3.1 Introduction

A gap in the current state-of-the-art was identified for a BEM that can simulate reality sufficiently well to provide actionable insights while still allowing for direct calibration in a scalable way. The BEM described in this chapter aims to fill this gap: a BEM that more closely resembles established engineering-based simulation tools in terms of detail and potential to provide insights about the actual energy performance of the building on the level of detail visualized in Figure 1.4, but can still be directly calibrated (without an intermediate em-ulator step) in a similar manner to the simpler 2–3-state gray-box models that are common in the literature.

Figure 3.1 shows a schematic of an example of the targeted district-heating multifamily building type, consisting of a substation serving a single or mul-tiple building(s) with both space heating and heat for DHW use and including heat losses in the DHW circulation and an eventual local pipe network. The aim of the BEM described in this chapter is to model everything behind the heat meter, so that the heat meter readings can be used to calibrate the model. The entire code is implemented in the statistical modeling language Stan [46], which with its automatic differentiation capabilities enables use of the devel-oped BEM for derivative-based inference, as later demonstrated in Chapter 4.

The developed BEM consist of a 14-node thermal network (that is, a system of 14 equations) which is a lumped and simplified version of the ISO 1:2017 [47] standard and was therefore named ISO14N. The ISO 52016-1:2017 [47] standard, which replaces the now deprecated ISO 13790:2008 [48] standard, presents a set of calculation methods for a building’s energy needs and internal air temperatures. The hourly method described in the standard proposes a system of linear equations that model heat transfer through opaque and transparent components of the envelope and air exchange between the in-ternal and exin-ternal environments. The calculation produces hourly inin-ternal air

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CRTL CRTL Heat meter θe Flow meter DH return DH supply Pipe network Cold water supply Radiators Taps Substation θset θset Domestic hot water circulation Radiators Taps θset

Figure 3.1.A district heating substation serving multiple buildings.

and component temperatures and heating and cooling loads. Each construc-tion component (for example, roof, windows and walls) is modeled as serially connected resistance and capacitance (RC) thermal networks. Compared with the deprecated ISO 13790:2008 [48] standard, the new hourly method more closely resembles established whole-building simulation software.

3.2 Thermal network

For Paper IV a dynamic BEM consisting of 14 node thermal network was developed; a lumped and simplified version of the ISO 52016-1:2017 [47] standard. It was originally presented in the same format as used in the ISO 52016-1:2017 [47] standard; a linearly solvable system of linear equations. Here, it is expressed as a continuous-time state-space model

dxt

dt = Atxt+ Btut (3.1)

where x ∈ R14×1, A ∈ R14×14, B ∈ R14×9, and u ∈ R9×1are the states (the temperature nodes of the thermal network), continuous-time transition matrix, continuous-time input coefficient matrix, and input data vector of the ISO14N thermal model, respectively, and the t subscript denotes time. The system 3.1 can be simulated directly in continuous-time (as shown in Paper IV, or by first transforming it into discrete-time format (see Section 4.1.1) which is more useful format for the probabilistic calibration used in Chapter 4.

Figure 3.2 visualizes the thermal network as a RC-network. All external walls (ew) are modeled as a lumped 3-node element, all window glazing (gl) are modeled as a lumped 2-node element, and the external roof (r f ) and the ground floor (g f ) are modeled with 3-node elements. The remaining internal mass im (internal walls, intermediate floors, and adiabatic external walls) are

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External walls (ew) κ1 κ2 κ3 h1 h2 θ2 θ3 Hve Htb Glazing (gl) h1 hsi Φsol θ2 Heating system κ1 κ2 κ3 h1 h2 θ2 θ3 Roof (rf) κ3 κ2 κ1 h2 h1 hsi hse θ2 θ1 Ground floor (gf) θ3 θgr Internal mass (im) κ2 κ1 h1 θ1 Cint θint hse θ1 θ1 θ1 hsi Internal gains Heat transfer Radiative heat transfer θ2 hse hse Φint Φhyd κ1 κ2 hsi hsi θsky θe

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A = −1·                                hse+h1 –h1 0 0 0 0 0 0 0 0 0 0 0 0 –h1 h1+h2 –h2 0 0 0 0 0 0 0 0 0 0 0 0 –h2 h2+hsi 0 0 –f·hri 0 –f·hri 0 –f·hri 0 0 –f·hri–hci 0 0 0 hse+h1 –h1 0 0 0 0 0 0 0 0 0 0 0 0 –h₁ h₁+h₂ –h₂ 0 0 0 0 0 0 0 0 0 0 –f·hri 0 –h2 h2+hsi 0 –f·hri 0 –f·hri 0 0 –f·hri–hci 0 0 0 0 0 0 hse+h1 –h1 0 0 0 0 0 0 0 0 –f·hri 0 0 –f·hri –h1 h1+hsi 0 –f·hri 0 0 –f·hri–hci 0 0 0 0 0 0 0 0 h1 –h1 0 0 0 0 0 0 –f·hri 0 0 –f·hri 0 –f·hri–h1h1+hsi 0 0 –f·hri–hci 0 0 0 0 0 0 0 0 0 0 hse+h1 –h1 0 0 0 0 0 0 0 0 0 0 0 0 –h₁ h₁+h₂ –h₂ 0 0 0 –f·hri 0 0 –f·hri 0 –f·hri 0 –f·hri 0 –h2 h2+hsi–hci 0 0 –r· hci 0 0 –r· hci 0 –r· hci 0 –r· hci 0 0 –r· hci ∗                                h1/κ11/κ21/κ31/κ11/κ21/κ31/κ11/κ21/κ11/κ21/κ11/κ21/κ31/Cint i> ×14 ∗_r_{· h} ci+r· hci+r· hci+r· hci+r· hci+ Htb

u =h θe θsky φsol φint φhyd θgrIhorIverφve

i B =                               

hce+(1−Fsky)hreFskyhre 0 0 0 0αsol 0 0

0 0 0 0 0 0 0 0 0

0 0 fr;sol/ ∑ r fr;int/ ∑ r fr;hyd/ ∑ r 0 0 0 0

hce+(1−Fsky)hreFskyhre 0 0 0 0 0 αsol0

0 0 0 0 0 0 0 0 0

hce+(1−Fsky)hreFskyhre 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 hse 0 0 0

0 0 0 0 0 0 0 0 0

Htb 0 fc;sol fc;int fc;hyd 0 0 0 1

                                                              1/κ1 1/κ2 1/κ3 1/κ1 1/κ2 1/κ3 1/κ1 1/κ2 1/κ1 1/κ2 1/κ1 1/κ2 1/κ3 1/Cint                                ×14 Figure 3.3.The state-space matrices of the thermal network. Colors indicate building element: roof (rf),external walls (ew),glazing (gl),internal mass (im)andground

floor (gf). denotes matrix element-wise multiplication and ∑ r is the sum of the

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represented with a 2-node element. Thus, the thermal network consists of 14 unknown temperature nodes (including the internal air node θint).

Figure 3.3 shows how the state-space matrices are constructed. Parameters are colored according to which building element they belong to (i.e.h1in place

of h1;r f). The purpose of the coloring is to better visualize how the coupling

between the building elements, external environment, and the internal air node are constructed. The states of the elements are all kept normalized to per square meter surface, while the internal node is normalized to per square meter floor.

3.2.1 Ratios and fractions

Geometrical data are given as ratios (r) between the total interior surface area of each building element type and the total floor area (unit ms/mfl). These ratios can be though of as weighting factors describing how large an impact one type of building element has on the average thermal balance of the whole building. The ratios are estimated as follows

rr f = rg f = 1/Nf l

rew= P · Hf l· Nf l/Af l− rgl rim= (2 − 2/Nf l) + 1.5

(3.2)

where Nf l is the number of floors of the building, Af l is the total floor area of the building, P is the building perimeter, and Hf l is the average internal floor height. The term (2 − 2/Nf l) in rimcalculates the surface area of internal floors and ceilings per total floor area and will result in a value between 0 and 2, depending on the number of floors. The additional 1.5 constant represents the internal walls, and it is based on the obsolete ISO 13790:2008 [48] stan-dard (where the ratio total interior surface area to floor area is given as 4.5). The glazing-to-floor area ratio rgl is a property that is often regulated in stan-dards and building codes, typically ranging between 0.1 and 0.2 for Swedish multi-family buildings. The surface area fractions f are simply calculated by dividing the ratio (r) of each building element with the sum of all ratios of the five building element types.

The heat input from solar heat gains (φsol), internal heat gains (φint), and the hydronic heating system (φhyd) are divided into a radiative and convective fractions. fris the fraction of radiative heat and fcis the fraction of convective heat ( fr= 1 − fc). Default convective fractions from the ISO 52016-1:2017 [47] standard are as follows fc;sol= 0.1, fc;int = 0.4, and fc;hyd= 0.4.

The radiative heat transfer between the exterior surfaces and the the sky

and surroundings depends on the sky view factors Fsky. For a non-shaded

horizontal roof the sky view factor is 1.0, and for non-shaded vertical elements (external walls and window glazing) the sky view factor is 0.5.

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3.2.2 Heat transfer coefficients

The surface interior heat transfer coefficients are given as

h_si;el= hri+ hci;el (3.3)

where hri is the surface interior radiative heat transfer coefficient (assumed constant at 5.13, the radiative heat transfer between two surfaces at 20◦C and 0.9 emissivity) and hci;el is the surface interior convective heat transfer coef-ficient, and the subscript el denotes one of the five building elements. The

hci constants are taken from ISO 52016-1:2017 [47] (Table 25) and describe

conventional convective transfer coefficients for interior surfaces oriented up-wards, horizontally, or downward:

hci;r f = 0.7, hci;ew= hci;gl= hci;im= 2.5, hci;g f = 5.0 (3.4)

The surface exterior heat transfer coefficients are calculated as follow hse;r f ;t= hre+ hce;r f ;t, hse;ew;t = hre+ hce;ew;t

h_se;gl;t= hre+ hce;gl;t, hse;g f = hgr;vi

(3.5) where hre is the surface exterior radiative heat transfer coefficient (assumed constant at 4.14, the radiative heat transfer between two surfaces at 0◦C and 0.9 emissivity), hce;el;tis the time-varying exterior surface convective heat transfer coefficients (see Section 3.5.5), and hgr;vi is the heat transfer coefficient for a virtual ground layer calculated according to ISO 13370:2017 [49].

In the ISO 52016-1:2017 [47] standard, opaque building elements are by default modeled with 5-nodes and window glazing with 2-node elements. The distribution of the thermal resistances are slightly weighted towards the center of the building elements, while heat capacity nodes are weighted de-pending on chosen class (mass concentrated externally, internally, inside, or equally). In this implementation, 3-node elements are used for the opaque el-ements; therefore, there is less flexibility in distributing the thermal properties over the elements. The thermal resistance is equally distributed

h_{1;r f} = h2;r f = 2/Rr f, h1;gl= 1/Rgl

h1;ew= h2;ew= 2/Rew, h1;im= 1/Rim

h1;g f = 1/(Rg f/2 + Rgr), h2;g f = 2/Rg f

(3.6)

where Rgrrepresent the thermal resistance a 0.5 m thick soil layer (0.25 m2s· K/W) and the thermal resistances for the other elements are calculated as

Rel= 1/Uel− 1/(hsi;el) − 1/hse;el (3.7)