A probabilistic framework towards metamodeling the

ARENBERG DOCTORAL SCHOOL Faculty of Engineering Science

A probabilistic framework towards metamodeling the impact of residential heat pumps and PV on low-voltage grids

Christina Protopapadaki Supervisor: Prof. dr. ir.-arch. Dirk Saelens

Dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Engineering Science (PhD): Civil Engineering December 2018

A probabilistic framework towards metamodeling the impact of residential heat pumps and PV on low-voltage grids Christina PROTOPAPADAKI

A probabilistic framework towards metamodeling the impact of residential heat pumps and PV on low-voltage grids Christina PROTOPAPADAKI

Examination committee: Prof. dr. ir.-arch. Dirk Saelens, supervisor Prof. dr. ir. Jean Berlamont, chair preliminary defense Prof. dr. ir. Robert Puers, chair public defense Prof. dr. ir. Johan Driesen Prof. dr. ir. Lieve Helsen Prof. dr. ir.-arch. Staf Roels Prof. dr. ir. Vincent Lemort, Université de Liège Prof. dr. ir. Jessen Page, HES-SO Valais-Wallis

Dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Engineering Science (PhD): Civil Engineering

December 2018

© 2018 KU Leuven – Faculty of Engineering Science Uitgegeven in eigen beheer, Christina Protopapadaki, Kasteelpark Arenberg 40 - box 2447, B-3001 Leuven (Belgium)

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I don’t pretend we have all the answers. But the questions are certainly worth thinking about. – Arthur Clarke

Acknowledgments Research can be exciting, gratifying and even fun. At times, though, it is frustrating and tiring. Throughout this journey, there have been people around me to offer guidance, share good moments and support me in less pleasant times. To those I would like to address a word of acknowledgment. My deepest gratitude goes to my thesis supervisor, who’s always been there to advise and support me through this adventure. Dirk, I greatly appreciate the freedom and trust you offered me to engage in research beyond disciplinary boundaries. I appreciate the advice and guidance you gave me when I got stuck, the time you spent in endless meetings discussing the very purpose of my work. Thank you for those times you calmed me down when things went wrong. Your encouragement and appreciation kept me going. This work is the result of a rather lonely ride. Therefore, I’m particularly grateful for all collaborations, the small interesting discussions and useful input I got from colleagues in Building Physics and TME. Ruben and Glenn, you gave me a great boost at the start of this PhD, and you were there to answer my questions and resolve my doubts. You’ve taught me a lot, thank you! TME guys, thanks for the Modelica help and teamwork organizing the crash course. GOA colleagues, I really enjoyed our collaboration in the beginning of my PhD, but I wish we had more time to realize all the great ideas we discussed together! At the other end of this PhD, colleagues from the SBO project, thanks for your patience this last year, where I have been entirely absorbed by this thesis writing. I’ll make up to you in the coming years. Vincent, you get all credits for the figures in this thesis. You should have also taught me Python before leaving! Tianfeng, thanks for all the geeky help on mathematics. Hans, your door was always open for questions and your willingness to offer assistance seems inexhaustible. Thank you for all the discussions and advice on statistics; they’ve helped me a great deal. A warm thank you also to the members of my examination committee, Staf, Lieve, Johan, Vincent and Jessen, for the time they dedicated to my thesis, the comments and interesting suggestions for improvement they provided.

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Acknowledgments

These five years of research wouldn’t have resulted in this dissertation, hasn’t it been for the great colleagues and friends, making it worthwhile. A day didn’t pass at the office without a laugh, at coffee break, Alma, sandwich lunch, or impromptu breaks. Conferences and Annex meetings were always fun to go to. Outside office hours, we also shared memorable moments: trips, Christmas dinners, BBQs, parties, game nights, spicy Chinese dinners, laser tag at the office! Thank you all for the amazing time! I will not attempt to list all names of people I shared good memories with, for fear of leaving someone out. Nevertheless, I have some special thanks for some of you. Dong Hui, Maciej, you guys were the greatest housemates and board game partners; you made my coming back home so much more pleasant. Μάνθο and Μήτσο, thank you for keeping my Greek side alive! I’m open for a debate on this statement; I always enjoy arguing with you! Jelena, I couldn’t be happier to have shared my office and life in Leuven with you. You’ve been an example of the most hardworking person, yet full of energy and always ready for adventure. Juliana and Ina, you’ve saved me so much time, taking up my teaching tasks and giving me priority for simulations. Thank you so much for your thoughtfulness! Ando, Wouter and Fabio, these last months of thesis writing would have been much worse without you. Especially the weekends! Fabio, you haven’t stopped supporting me, therefore, I also say to you: A bit more to go. Dai! Ruben, you were my go-to colleague at work, and also my go-to person at home. I couldn’t thank you more, for your work-related help, and for your patience and support at home. I will not forget the dinners you delivered at the office so I could work till late on my manuscript. Little brother, Γιώργο, you’ve heard all my nagging, kept me company when I needed a break, and shared the best music to keep me going. Thanks! Σοφία, sister, you’re not always there, but when you are, you’re an inspiration. Thank you for giving me a different perspective on life and on what really matters. Mom, there is no time I called you frustrated from work, and you didn’t convince me everything would be fine. Everything is finally fine. Thank you for being the best mom! Last, I’d like to thank my dad, whom I miss dearly, for teaching me the importance of being fair, humble and kind. Dad, I hope this dissertation would have made you proud.

Christina, December 4, 2018

Abstract European policies aiming to reduce residential energy use and carbon emissions promote energy efficiency in buildings, heating electrification and renewable electricity generation. Low-carbon technologies, such as heat pumps and rooftop photovoltaic systems (PV) can significantly affect electricity load patterns, with potential impacts on all levels of the electricity system. Technical constraints at the distribution grid level may limit the potential penetration of these technologies in practice, as it would require load control, specialized equipment, storage or grid reinforcement, at considerable cost. These issues, however, are often ignored or simplistically considered in high-level assessments. Therefore, this thesis aims to describe and quantify, in a probabilistic way, the impact of residential heat pumps and PV on low-voltage distribution grids. It furthermore aims to explore metamodeling as a method to allow a computationally inexpensive assessment of this impact. In this way, the results could be easily utilized in high-level studies. First, a framework is developed, that allows to probabilistically analyze the lowvoltage grid impacts of heat pumps and PV, on multiple different network cases. The methodology combines detailed simulation of buildings and networks using Modelica, and a Monte Carlo approach to incorporate uncertainties with respect to the loads. The influence of several modeling and methodological assumptions is furthermore investigated. The framework is then employed to study the impact of heat pumps and PV on Belgian low-voltage grids. The results demonstrate the problems arising from widespread installation of these technologies, and identify the main influential grid and building parameters. Sensitivity analyses additionally reveal the importance of boundary conditions, such as the weather, in such assessments. In order to translate these findings into decision support tools for large-scale studies, the second part of this thesis focuses on metamodeling grid impact indicators. Metamodels can approximate results of the detailed but computationally expensive model, using few important inputs and requiring much less prediction time. Therefore, a metamodeling methodology is presented, which synthesizes

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Abstract

common approaches found in literature and applies them to the specific case of grid impact analysis. Each step of the metamodeling process is discussed in detail, highlighting challenges related to this problem. The methodology is then applied to train metamodels for grid impact indicators, based on grid parameters and some building properties. Simple models, namely ridge and logistic regression, are first explored, discussing model checking, inference, model selection, and the specific needs of different indicators. Neural networks are then developed for the minimum voltage indicators, since these are more difficult to predict with simple regression. Several modeling choices are investigated for these models, in order to highlight the complexity of metamodeling, but also the potentials. Promising performance was found for both the linear models and neural networks. However, the latter require significantly more training time. As such, their use could be reserved for cases where the linear models do not provide satisfactory results. Overall, this research would suggest that metamodeling has the potential to help incorporate local grid restrictions in large-scale assessments, for instance policy evaluation, urban planning or network management and design.

Beknopte samenvatting Een probabilistische methode voor het meta-modelleren van de impact van residentiële warmtepompen en PV op laagspanningsnetten Europees beleid, gericht op het verminderen van het energiegebruik in woningen en CO2-emissies, bevordert de energie-efficiëntie in gebouwen, de elektrificatie van verwarming en de opwekking van hernieuwbare elektriciteit. Lage-emissie technologieën zoals warmtepompen en fotovoltaïsche (PV) systemen op het dak kunnen het elektriciteitsprofiel sterk beïnvloeden. Dit heeft een potentiële impact op alle niveaus van het elektriciteitssysteem. Technische beperkingen van het distributienet limiteren de mogelijke uitrol van deze technologieën in de praktijk omdat de zware elektrische belasting specifieke oplossingen zoals opslag of netversterkingen zouden vereisen. Deze beperkingen worden echter vaak genegeerd of op een te vereenvoudigde manier bekeken in high-level beoordelingen. Daarom is dit proefschrift gericht op het probabilistisch beschrijven en kwantificeren van de impact van residentiële warmtepompen en PV op laagspanningsdistributienetten. Verder betracht het om meta-modellering te verkennen als numeriek minder veeleisende methode voor een vereenvoudigde beoordeling wat het gebruik van de resultaten in high-level studies moet faciliteren. Eerst wordt een methode ontwikkeld die het mogelijk maakt om de effecten van warmtepompen en PV op verschillende cases van laagspanningsdistributienetten te analyseren. Deze methodologie combineert een gedetailleerde simulatie van gebouwen en netwerken door middel van Modelica en een Monte Carlo-benadering om onzekerheden met betrekking tot de elektrische belasting in te rekenen. De invloed van modellen en methodologische aannames wordt verder geanalyseerd en vervolgens wordt de methode gebruikt voor de studie van de impact van warmtepompen en PV op Belgische laagspanningsdistributienetten. De resultaten illustreren de problemen die optreden bij een doorgedreven implementatie van deze technologieën en identificeren de belangrijkste netwerken gebouwparameters. Sensitiviteitsanalyses onthullen bovendien het belang van randvoorwaarden, zoals de weersomstandigheden, bij dergelijke beoordelingen.

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Beknopte samenvatting

Om deze bevindingen vervolgens te kunnen vertalen in beslissingsondersteunende tools voor grootschalige onderzoeken, richt het tweede deel van dit proefschrift zich op het ontwikkelen van meta-modellen. Hiermee kan de beoordeling van grid-impact indicatoren uit de complexe simulaties met behulp van een aantal goedgekozen inputparameters op een numeriek efficiënte manier benaderd worden. De methodologie voor meta-modellering is gebaseerd op de relevante literatuur terzake en past deze toe op het specifieke geval van grid-impactanalyse. Elke stap in de meta-modellering wordt uitvoerig besproken, met de nadruk op specifieke uitdagingen en wordt vervolgens toegepast om meta-modellen te trainen voor gridimpactindicatoren op basis van gridparameters en enkele gebouweigenschappen. Eenvoudige modellen, zijnde ridge regressie en logistische regressie, worden eerst verkend waarbij modelcontrole, inferentie, modelselectie, en de specifieke behoeften van verschillende indicatoren worden besproken. Vervolgens worden neurale netwerken ontwikkeld voor indicatoren die de minimale spanning analyseren. Deze bleken immers moeilijker te voorspellen met een eenvoudige regressie. Verschillende modelleringskeuzes worden voor deze modellen onderzocht om de complexiteit en opportuniteiten van meta-modellen te benadrukken. Zowel de lineaire modellen als de neurale netwerken gaven veelbelovende prestaties. Omdat deze laatste methode echter aanzienlijk meer trainingstijd vereist, wordt aanbevolen om het gebruik ervan voor te behouden voor gevallen waarin de lineaire modellen geen bevredigende resultaten opleveren. Algemeen gesteld, toont dit onderzoek aan dat meta-modellering het potentieel heeft om lokale netwerkbeperkingen op te nemen in high-level beoordelingen, bijvoorbeeld bij beleidsevaluaties, stadsplanning of netwerkbeheer en -ontwerp.

Contents

Acknowledgments

i

Abstract

iii

Beknopte samenvatting

v

Contents

vii

List of Figures

xi

List of Tables

xv

1

Introduction

1

1.1

Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Objectives, research scope and contributions . . . . . . . . . . . . . .

3

1.3

Methodology and thesis structure . . . . . . . . . . . . . . . . . . . . .

5

2 Background and state-of-the-art

7

2.1

Decarbonizing the residential sector to mitigate climate change . .

7

2.2

Heat pumps and PV: Potential and challenges . . . . . . . . . . . . .

9

2.3

Review of literature on the assessment of LCTs . . . . . . . . . . . . .

11

2.3.1

11

The need to consider grid interactions . . . . . . . . . . . . .

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2.4

2.5

Contents

2.3.2

Grid impact assessment . . . . . . . . . . . . . . . . . . . . . .

13

2.3.3

State-of-the-art: Recommendations and limitations . . . . .

16

Metamodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4.1

What is metamodeling? . . . . . . . . . . . . . . . . . . . . . .

18

2.4.2

Review of metamodeling in energy-related applications . . .

19

2.4.3

Potential of metamodeling for grid impact assessment . . .

20

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3 Framework for probabilistic grid impact analysis

23

3.1

General methodology and assumptions . . . . . . . . . . . . . . . . .

24

3.2

Quantities of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.3

Inputs and experimental design . . . . . . . . . . . . . . . . . . . . . .

30

3.3.1

Factors influencing low-voltage grid impact indicators . . .

31

3.3.2

Input for probabilistic framework and metamodel . . . . . .

33

3.3.3

Experimental design . . . . . . . . . . . . . . . . . . . . . . . .

36

Simulation models and procedures . . . . . . . . . . . . . . . . . . . .

39

3.4.1

Building stock . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.4.2

Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Sensitivity to modeling assumptions . . . . . . . . . . . . . . . . . . .

55

3.5.1

Distribution island modeling approach . . . . . . . . . . . . .

56

3.5.2

Input data and simulation temporal resolution . . . . . . . .

59

3.5.3

Number of replications . . . . . . . . . . . . . . . . . . . . . .

62

3.5.4

Number of design points . . . . . . . . . . . . . . . . . . . . .

69

3.6

Strengths and limitations of the methodology . . . . . . . . . . . . .

73

3.7

Probabilistic framework: Recapitulation . . . . . . . . . . . . . . . . .

74

3.4

3.5

4 Impact of heat pumps and PVs on Belgian residential feeders 4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75

Contents

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ix

4.2

Grid parameter sensitivity . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3

Load profile analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.4

Cable and transformer overloading . . . . . . . . . . . . . . . . . . . .

89

4.5

Voltage problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.6

The role of building properties . . . . . . . . . . . . . . . . . . . . . .

97

4.7

Weather scenario analysis and simulation period . . . . . . . . . . . 100

4.8

Heat pump power factor scenario analysis . . . . . . . . . . . . . . . 113

4.9

Households composition scenario analysis . . . . . . . . . . . . . . . 116

4.10 Grid impact: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Metamodeling grid impact indicators: Methodology

123

5.1

Overview of metamodeling methodology . . . . . . . . . . . . . . . . 123

5.2

Metamodel goal: inference vs. prediction . . . . . . . . . . . . . . . . 124

5.3

Selection of output variables . . . . . . . . . . . . . . . . . . . . . . . . 125

5.4

Definition of metamodel features . . . . . . . . . . . . . . . . . . . . . 128

5.5

Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.6

Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.7

Metamodeling techniques . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.8

Metamodel training and validation . . . . . . . . . . . . . . . . . . . . 145

5.9

Metamodeling methodology: Recapitulation . . . . . . . . . . . . . . 148

6 Metamodeling grid impact indicators: Application

151

6.1

Predictors and response variables . . . . . . . . . . . . . . . . . . . . . 151

6.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3

Linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.3.1

Linear regression with regularization: Ridge regression . . . 158

6.3.2

Peak demand: analysis of ridge regression model . . . . . . . 160

6.3.3

Annual net consumption: mean vs. all replications . . . . . . 166

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6.4

6.5

6.6

Contents

6.3.4

Minimum voltage: response transformation . . . . . . . . . . 171

6.3.5

Logistic regression model . . . . . . . . . . . . . . . . . . . . . 173

6.3.6

Voltage violation detection with logistic regression . . . . . . 175

Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.4.1

Neural network models . . . . . . . . . . . . . . . . . . . . . . 179

6.4.2

Minimum voltage: mean and p 10 . . . . . . . . . . . . . . . . 181

6.4.3

Minimum voltage: 20 vs. 50 replications . . . . . . . . . . . . 183

6.4.4

Minimum voltage: number of design points . . . . . . . . . . 184

6.4.5

Minimum voltage: p 10 vs. all replications . . . . . . . . . . . 186

6.4.6

Voltage violation detection with neural network . . . . . . . 188

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.5.1

Linear models vs. Neural Networks . . . . . . . . . . . . . . . 189

6.5.2

Potential of metamodels as decision support tools . . . . . . 191

Metamodeling application: Conclusion . . . . . . . . . . . . . . . . . 194

7 Conclusion

199

7.1

Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.2

Summary of methodology and conclusions . . . . . . . . . . . . . . . 200

7.3

Recommendations for improvements and further research . . . . . . 205

List of Abbreviations

209

List of Symbols

211

Bibliography

219

List of publications

233

List of Figures

1.1

Schematic representation of electricity supply chain. . . . . . . . . . . . . . . . . . . . . .

2

1.2

Outline of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1

Schematic representation of the relation between the real physical system, the simulation model and metamodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1

Main steps of the methodology for probabilistic impact analysis and metamodeling. . . 24

3.2

Overview of the simulation experimental design and the resulting dataset for metamodeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3

Creation of a building stock, consisting of different load profiles. . . . . . . . . . . . . . . 40

3.4

Geometric model for the buildings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5

Building simulation results for example building, showing temperature and electrical loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6

Comparison of day-zone temperatures with and without back-up heating for moderately and well insulated dwelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.7

Rural and urban feeders with dummy representation of the remaining distribution island. 52

3.8

Building stock subsets used to sample buildings of a given construction quality Q. To define the subsets, buildings are ordered based on Uavg . . . . . . . . . . . . . . . . . . . . 54

3.9

Modeling approaches investigated for the distribution island simulation. . . . . . . . . . 56

3.10

Differences in per unit current and voltage indicators for single and dummy island approaches, compared to full island simulations, per transformer rated capacity S n . . . 57

3.11

Differences in Umin for single feeder and dummy island approaches compared to island simulation, per number of buildings NI (left) and heat pump penetration level r HP,I (right) in the remainder island. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.12

CPU-time for annual simulations with the different modeling approaches. . . . . . . . . 59

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List of Figures

3.13

Differences in 10-min averaged minimum and maximum voltage per resolution ∆t , with respect to 1-min resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.14

Median differences in maximum feeder current and transformer peak power with respect to 1-min resolution, per resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.15

CPU-time for annual simulations and different temporal resolution. . . . . . . . . . . . 62

3.16

Bootstrap relative standard deviations for the sample mean of the annual feeder electricity consumption, per number of replications. . . . . . . . . . . . . . . . . . . . . . 64

3.17

Bootstrap relative standard deviations for the feeder peak electricity demand, for all design points, per number of replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.18

Distribution of the normalized feeder peak demand over all design points, per number of replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.19

Bootstrap standard error for the sample mean and percentile of grid indicators, per number of replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.20

Distribution of the minimum voltage over an example selection of cases, per number of replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.21

Bootstrap standard error of grid indicators, per number of design points, for the dataset mean and a percentile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.22

Distribution of the normalized mean E t,d,net and the 10th percentile of the minimum voltage Umin over an example selection of cases, per number of design points in the whole dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1

Time-series annual results for example case. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2

Probability density functions of grid impact indicators, based on kernel density estimation. 78

4.3

Sensitivity of grid impact indicators to the grid parameters. . . . . . . . . . . . . . . . . . 83

4.4

Example load-duration curves for rural neighborhoods of N = 30 buildings and varying levels of heat pump and PV penetration, as well as construction quality (see Uavg ). . . . 86

4.5

Simultaneity factors k s for different types of loads. . . . . . . . . . . . . . . . . . . . . . . 88

4.6

Distribution of maximum transformer apparent power S max for an example feeder case based on 20 replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7

Proportion of grid cases with a probability of transformer or cable overloading equal or inferior to P, depending on heat pump penetration levels of the distribution island or feeder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8

Contour plots of the fitted 90th percentile of the transformer maximum load S max , on the distribution island heat pump penetration rate and number of buildings, for different neighborhood types and transformer rated capacities. . . . . . . . . . . . . . . 92

4.9

Contour plots of the fitted 90th percentile of the cable maximum current I max , on the feeder heat pump penetration rate and number of buildings, for different neighborhood types and building construction quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

List of Figures

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xiii

4.10

Proportion of grid cases with a probability of voltage violation equal or inferior to P, depending on heat pump or PV penetration levels in the feeder. . . . . . . . . . . . . . . 95

4.11

Contour plots of the fitted 10th percentile of the minimum voltage Umin , on the feeder heat pump penetration rate and number of buildings, per neighborhood type and transformer nominal voltage Uref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.12

Spearman’s rank partial correlation coefficients between indicators and different building-related parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.13

Top: Monthly mean ambient temperature and global horizontal irradiation for all scenarios. Bottom: difference compared to scenario 1-Avg. . . . . . . . . . . . . . . . . . 102

4.14

Change in transformer annual energy demand and backfeeding for different weather scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.15

Change in overloading and voltage indicators for different weather scenarios. . . . . . . 105

4.16

Percentage of overloaded transformers and feeders for different weather scenarios. . . . 106

4.17

Percentage of feeders with voltage violations for different weather scenarios. . . . . . . 107

4.18

Selection of representative week W for Umin and P t,bf , based on different criteria, for scenario 1-Avg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.19

Percentage of feeders with voltage violations for the annual simulation or the representative week, chosen based on ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.20

Differences in indicators for lower heat pump power factor cos φ compared to reference cos φ = 1, as function of heat pump penetration level in the feeder r HP (or the entire island r HP,t for S max ). Lines denote the median and shaded areas contain 90 % of data points, from percentile p 5 to p 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.21

Comparison of probability density functions of grid impact indicators, for single-person household ratio of 35 % and 20 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.1

General metamodeling methodology and basic steps. . . . . . . . . . . . . . . . . . . . . 124

5.2

Distribution of Umin for an example feeder case based on 20 replications. . . . . . . . . 126

5.3

Distribution of P viol over all 1296 design points. . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4

Replica of Figure 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.5

Value assigned to variable x q for design point i , based on the sampled probability for its cumulative distribution function F (x q ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.6

Graphical representation of a multilayer perceptron feed-forward neural network with one hidden layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.7

Hyperbolic tangent sigmoid function used in hidden layer neurons of the neural network.138

5.8

Principle of support vector regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.9

Overview of metamodel training and validation general procedure. . . . . . . . . . . . . 146

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List of Figures

6.1

Distribution of mean minimum voltage over all 1296 design points of dataset D, and split by level of reference voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2

Performance on test set of the final ridge model for p 90 of P d . . . . . . . . . . . . . . . . 161

6.3

Residual analysis for ridge regression for p 90 of P d . . . . . . . . . . . . . . . . . . . . . . . 162

6.4

Cross-validation MSE and ridge trace plot for regression model for p 90 of P d . . . . . . . 163

6.5

Slices of the fitted response surface from the LR model for p 90 of P d . . . . . . . . . . . . 165

6.6

Performance on test set of the final ridge model for E t,d,net . . . . . . . . . . . . . . . . . . 167

6.7

Relative prediction error on test set of the final ridge model for E t,d,net . . . . . . . . . . . 168

6.8

Residuals on test set of the final ridge models for E t,d,net of individual replications, with Q or Q des as predictors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.9

Performance on test set of the final ridge model for U min . . . . . . . . . . . . . . . . . . . 171

6.10

Distribution of the log-transformed difference Uref −U min , with a normal distribution fit, based on all 1296 design points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.11

Test residuals of the final log-linear ridge regression model for U min . . . . . . . . . . . . 173

6.12

Test precision and recall against cutoff, for the final selected model of all four classification modeling approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.13

Performance on test set of the final NN model for U min . . . . . . . . . . . . . . . . . . . . 182

6.14

Performance on test set of the final NN model for p 10 of Umin . . . . . . . . . . . . . . . . 182

6.15

Test residuals of the final NN model for p 10 of Umin based on 50 replications. . . . . . . 184

6.16

Comparison of NN model performance for different numbers of design points. . . . . . 185

6.17

Test residuals of the final NN model for Umin of individual replications, with features Q des for feeder and rest of island buildings. . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.18

Average training time required for one inner CV iteration for each of the different models used in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.19

Example probability of voltage violation plots based on prediction from logistic regression model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.20

Example contour plot and 3D surface plot of predicted minimum voltage with NN. . . . 193

List of Tables

3.1

Overview of indicators used for grid impact analysis and metamodeling . . . . . . . . . 30

3.2

Inputs for the probabilistic framework, classified as grid parameters, uncertain factors or scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3

Overview of the varied building input parameters and their considered distributions. . 42

3.4

Window parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5

Inputs for the probabilistic framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6

Cable properties used in simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7

Transformer properties used in simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1

Percentage of overloaded transformers, per rated capacity. . . . . . . . . . . . . . . . . . 92

4.2

Percentage of overloaded feeders, per cable strength. . . . . . . . . . . . . . . . . . . . . . 93

4.3

Percentage of simulations with voltage violations, per reference transformer voltage Uref , cable type Ca, neighborhood type T and construction quality Q. . . . . . . . . . . . 95

4.4

Overview of the considered neighborhood building-related parameters and the calculation method used to define them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5

Weather scenarios with their temperature and irradiation settings, generated by Meteonorm 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6

Resulting selected representative week W (start day) for all indicators, weather scenarios and criteria. The ratio of cases (γ) falling in week Wγ is also given. The minimum and maximum values of γ per indicator are furthermore highlighted in bold. For definitions, see Equations 4.5 to 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.7

Percentage of voltage violations and overloading cases for different heat pump power factor cos φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.8

Percentage of voltage violations and overloading cases for the original dataset, with 35 % single-person households, and the scenarios with 30 and 20 %. . . . . . . . . . . . . . . . 119

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List of Tables

6.1

Metamodel predictors used for different response variables. . . . . . . . . . . . . . . . . 153

6.2

Cross-validation (CV) and test performance of ridge regression model for p 90 of P d . . . 160

6.3

Comparison of regression coefficients from ridge regression and linear regression for 3rd degree polynomial models for p 90 of P d . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4

Cross-validation and test performance of ridge regression model for E t,d,net . . . . . . . 167

6.5

Cross-validation and test performance of ridge regression model for E t,d,net of individual replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.6

Cross-validation and test performance of ridge regression model for E t,d,net of individual replications, using Q des as predictor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.7

Cross-validation and test performance of ridge regression model for U min . . . . . . . . 173

6.8

Cross-validation and test performance of the log-linear ridge regression model for U min .173

6.9

Cross-validation and test Brier scores based on calculation with Zviol and P viol , for the four different modeling approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.10

Cross-validation and test performance for the four different modeling approaches. . . . 178

6.11

Settings for neural networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.12

Cross-validation and test performance of NN model for U min and p 10 of Umin . . . . . . 181

6.13

Cross-validation and test performance of NN model for p 10 of Umin based on 50 replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.14

Cross-validation and test performance of NN model for Umin of individual replications, using Q des as features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.15

Predicted vs. actual percentage of cases where individual replications or p 10 per design point are found to violate the low voltage limit of 0.85 pu=195.5 V. . . . . . . . . . . . . . 188

6.16

Comparison of cross-validation and test performance for the NN classifier and the logistic regression models trained with stepwise selection. . . . . . . . . . . . . . . . . . . 189

1

Introduction What is the use of a house if you haven’t got a tolerable planet to put it on? – Henry David Thoreau

Humans have always sought progress and prosperity, finding solutions for the problems that emerged. Recent years have given rise to skepticism about whether the environment could sustain our ever growing needs. Climate change is high on the list of global issues, uniting many nations under this common concern. In order to reduce adverse impacts of climate change, our ingenuity should be devoted to crafting mitigation and adaptation solutions, with a view to establishing long term resilience and welfare. Both personal and collective actions are needed to that respect, starting from a change of conscience and moving on to policy and concrete implementation. This thesis contributes an effort towards the implementation of climate change mitigation solutions.

1.1

Problem statement

Buildings have an important role to play in the energy transition that has begun in the European Union and other countries. In order to adhere to recent commitments of the Paris Agreement, the European Union has adopted policies and regulations that promote energy efficiency and renewable energy supply. In Belgium, more stringent building performance regulations and subsidies for low-carbon technologies (LCTs) have prompted a rapid increase in the deployment of heat pumps and rooftop photovoltaic systems (PVs) in residential buildings. While heat pumps and PV contribute to the reduction of carbon emissions—a major driver of global warming and climate change—their wide deployment would also result in large impacts on the electricity grid. Residential heat pumps and PV are connected to the low-voltage (LV) distribution grid, which is the last link of the electricity supply chain between centralized production and residential

2

|

Introduction

Extra-High Voltage 220-380 kV

Transmission

Distribution

High Voltage

Medium Voltage

70-150 kV

Industrial consumer

1-36 kV

Solar plant Wind farm Commercial consumer Interconnection with other country

Heavy industry

Residential consumers

Wind farm

Feeder

Transformer Conventional generation Renewable generation Load

Low Voltage 0.4 kV

Distribution island

Figure 1.1: Schematic representation of electricity supply chain. This work focuses on residential LV distribution islands, representing several feeders connected to the same medium-to-low voltage (MV/LV) transformer, as indicated. consumption, as illustrated in Figure 1.1. Historically, LV distribution networks were designed for smaller loads, and were not meant to accommodate distributed generation, such as rooftop PVs, which create reverse power flow. Potential impacts of heat pumps and PVs on the electricity system, and specifically on the LV distribution grid include overloading and voltage quality issues, among others. A more detailed discussion of these challenges is given in Chapter 2. To allow substantial deployment of heat pumps and PV, measures should be taken to prevent the mentioned impacts. These measures may include grid reinforcements, specialized voltage control equipment, energy storage or smart control, all of which come with a cost. As a result, better understanding and quantification of the potential issues is essential for a correct appraisal of LCTs. Literature has begun to analyze the LV grid impacts of technologies such as heat pumps and PV, as reports Section 2.3.2 in detail. However, most studies limit their analysis to one or few specific networks. Furthermore, uncertainties in the loads are often not considered, while buildings characteristics are rarely taken into account in the representation of loads. Therefore, a framework is needed to allow probabilistic grid impact assessment for various networks, taking into account the contribution of building properties.

Objectives, research scope and contributions

|

3

An important disadvantage of such approaches is the large computation and data collection effort that they require. As a consequence, while they provide useful insights on potential grid impacts, they are not appropriate for use by non-experts with limited time, or when detailed data are not available or difficult to obtain. Therefore, there could be great value in developing methods to translate the findings of detailed probabilistic assessments into tools that can be used to support policy decisions, urban planning or network management.

1.2

Objectives, research scope and contributions

Based on the problem statement presented in the previous section, and further explained in Chapter 2, two main objectives can be defined for this work: Research objectives d Describe and quantify in a probabilistic way the impact of low-carbon solutions in residential buildings on LV distribution grids. d Explore metamodeling as a method to allow a computationally inexpensive assessment of these LV grid impacts. Some aspects of these two objectives need to be determined in more specific terms, refining the scope of this work, in order to develop an appropriate methodology. This thesis has its main focus on residential buildings, and in particular Belgian single-family dwellings. Therefore, the first objective deals with impacts created from low-carbon solutions implemented in those dwellings. Specifically, heat pumps and rooftop PV are the two considered technologies, for which increasing penetration levels are investigated. Both heat pumps and PV are seeing rapid growth in the Belgian residential sector [96, 227], and they represent the most common LCTs adopted to satisfy energy performance requirements [228]. Air-source heat pumps are chosen rather than geothermal ones, as they hold a greater market share in Europe and Belgium [227], but are also expected to have a larger impact on the grid because of their lower efficiency [12, 157]. Furthermore, micro-combined heat and power (CHP) systems were shown to have only small impact on the grid [158]. Other LCTs with potentially important impact on the grid are electric vehicles [158]. These were not included in the scope of this thesis, because they are not yet considered in building performance assessment. Nevertheless, the probabilistic framework is developed to allow incorporation of other technologies, including electric vehicles. Low-carbon solutions in buildings that do not directly interact with the electricity grid were not relevant in the context of this study, for instance biomass boilers or solar water heating.

4 |

Introduction

The impact of LCTs in residential buildings is analyzed for the LV distribution networks they are connected to. In the electricity grid overview of Figure 1.1, an example of a LV distribution island is depicted, which represents the level at which the impacts are considered in this work. A distribution island consists in several distribution feeders, all connected to the same medium-to-low voltage (MV/LV) transformer. The framework aims to evaluate technical issues for a variety of different Belgian distribution islands, with the same main configuration as depicted in Figure 1.1, but with varying size and different component ratings. The impacts examined in the framework relate to issues occurring in these distribution islands, under normal operation. These include mainly component overloading and voltage variations, evaluated on time-scales of few minutes. Impacts on the overall energy system are only considered in terms of peak and total annual demand required by the distribution island. More details on the potential impacts are given in Section 2.2, while the indicators used by the framework are described in Section 3.2. To estimate the impacts, this work considers a reference case where no measures have been yet taken to facilitate the integration of these LCTs. As such, the effectiveness of heat pump or PV adoption could be assessed taking into account that additional costs may arise for grid operation, reinforcements, voltage regulation or other measures. This thesis does not include financial calculations, but provides an estimate of when and if such measures would be required. The framework furthermore needs to analyze grid impacts in a probabilistic way, so that more robust estimates are made. This implies that uncertainties are taken into account in the calculation of grid impacts. Uncertainties considered in this work are related to building properties, occupants, and the location of loads on the grid. Last, the framework aims to describe and quantify these grid impacts. On the one hand, detailed analysis and simulation of time-series is needed, in order to understand in which way the grid is influenced, and which phenomena contribute to the observed impacts. On the other hand, quantification of the said impacts is necessary, by defining specific grid impact indicators, which can be used as proxies of the overall interaction of LCTs and the grid. The second objective pertains to the transition from the detailed probabilistic framework to simpler and faster tools for grid impact assessment, which can be utilized for policy support or the design of district energy systems. The specific method proposed in this thesis is metamodeling. Metamodels have been used in many engineering fields and energy-related topics to approximate real systems or more complex models, as explained in Section 2.4 and Chapter 5. Therefore, metamodeling could be a valuable method for the purpose of inexpensive grid impact assessment. This thesis aims to explore the potential of metamodeling in this context. That is, it aims to formulate an appropriate methodology for metamodeling grid impact

Methodology and thesis structure

|

5

indicators based on the probabilistic framework, identify challenges and specific requirements, and assess the performance of such developed metamodels. In this way, the outcome of this thesis is a methodology towards the creation of a metamodel-based grid impact assessment tool. Given the problem statement, which is based on the literature review presented in Chapter 2, and the mentioned objectives and scope, this thesis’ contributions are summarized below. Contributions d A methodology and framework for LV grid impact assessment of LCTs combining: ] ] ]

]

multiple different network cases, probabilistic representation of loads, explicit representation of buildings, such that scenarios with respect to building renovations can be assessed, and investigation of several methodological assumptions.

d Analysis of heat pump and PV impacts on Belgian LV feeders, where: ] ] ]

the potential impacts and main bottlenecks are quantified, the most influential grid and building parameters are identified, and sensitivity analyses on different boundary conditions are performed.

d A metamodeling methodology specially designed for grid impact indicators. d Exploration of metamodeling potential for grid impact assessment, where: ] ]

1.3

special requirements of different indicators are demonstrated, and the performance of certain metamodeling techniques is evaluated.

Methodology and thesis structure

With research scope and objectives defined in the previous section, the methodology and corresponding thesis structure can be summarized as illustrated in Figure 1.2. This figure gives an overview of the thesis outline, where chapters are represented as tiles of a puzzle, linked based on the dependencies between each of them. Chapter 2 first analyzes state-of-the-art literature on grid impact analyses and metamodeling, after giving a broader presentation of this work’s context. The literature review helped define the main objectives of this thesis, but also resulted in more specific requirements for the development of this thesis (see Section 2.3.3).

6

|

Introduction

Chapter 2

Chapter 5

Chapter 6

Background and state-of-the-art

Metamodeling methodology

Metamodeling application

Chapter 1

Chapter 3

Chapter 4

Chapter 7

Introduction

Probabilistic framework

Impact analysis on Belgian grids

Conclusion

Figure 1.2: Outline of the thesis. The puzzle connections represent links from one chapter to another. First, the probabilistic framework is developed in Chapter 3. This chapter discusses which indicators are of interest, how uncertain inputs are considered and how multiple grid cases are developed. It also describes the simulation models and assumptions in detail. Inputs for the proposed methodology come from the findings of Chapter 2, but also from the metamodeling methodology developed in Chapter 5. The latter influences aspects of the probabilistic framework, in particular with respect to the selection of grid cases to be examined. Chapter 4 then presents results from the probabilistic framework, in order to identify the main technical bottlenecks to the integration of heat pumps and PV in Belgian LV feeders. Furthermore, the contribution of different grid and building parameters on these results is examined, as well as the influence of assumptions on boundary conditions. In Chapter 5, an extensive review of metamodeling theory is performed, based on which the metamodeling methodology for grid impact indicators is developed. Due to lack of similar applications in literature, a synthesis is made, providing options and discussing challenges specific to metamodeling grid impact indicators. Among potential metamodeling techniques, linear models and neural networks are selected for detailed analysis and application in Chapter 6. These metamodels are trained for a selection of indicators resulting from the analysis in Chapter 4. The practical implementation of the metamodeling process is demonstrated in this chapter, emphasizing differences between indicators and analyzing the effect of several modeling choices. This chapter also provides an illustration of the benefits metamodels have to offer in grid impact assessment. Last, Chapter 7 provides a summary of the methodology and main conclusions, and it discusses limitations of the proposed framework as well as perspectives for improvement and continuation of this work.

2

Background & state-of-the-art

This chapter gives a more detailed introduction to the context for this research. First, low-carbon technologies (LCTs) are presented as solutions implemented in the residential sector for climate change mitigation. The potential of LCTs and challenges related to their integration in low-voltage (LV) grids are described, highlighting the need for evaluation of local grid restrictions. A state-of-the-art review is performed, which shows that large-scale studies often fail to consider those restrictions. On the other hand, the main findings and limitation of literature that examines grid impacts are identified, establishing the niche for this work and determining the main requirements for the proposed methodology. Last, metamodeling as a solution for quicker and easier assessment is introduced. The general approach is described, with mention to current applications in energy-related topics, after which the relevance and potential of grid impact metamodels are discussed.

2.1

Decarbonizing the residential sector to mitigate climate change

The consequences of climate change are already being felt around the globe, not only affecting physical and biological systems, but human populations as well. The Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) presented evidence of the observed changes and impacts on natural and human systems [102]. A changing climate could destabilize ecosystems, undermine food security, deteriorate human health, damage infrastructure, slow down economic growth and increase displacement of populations. The same report argues that it is extremely likely that anthropogenic drivers, and in particular greenhouse gas emissions, are a major cause of the observed climate change. Furthermore, abrupt and irreversible changes are more likely as global warming increases. Therefore, the report advises to pursue adaptation and mitigation strategies in order to build a sustainable and climate-resilient future.

8

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Acknowledging the eminence of climate change issues, the Paris Agreement was sealed in December 2015 between members of the United Nations Framework Convention on Climate Change, committing to limit the global average temperature increase well below 2 °C above pre-industrial levels. To fulfill this commitment, the European Commission presented in November 2016 the Clean Energy for All Europeans package of measures, that aims to provide the legislative framework to facilitate the clean energy transition. To achieve the target of 40 % reduction in CO2 emissions by 2030, compared to 1990 levels, these measures prioritize energy efficiency, renewable energy and consumer participation via a redesigned electricity market and smart metering [57]. In the European Union, buildings are responsible for about 40 % of energy use and 36 % of CO2 emissions [59], offering large potential for energy savings. In fact, the Energy Performance of Buildings Directive was recast to include proposals of the Clean Energy package in June 2018 [63]. Furthermore, a political agreement was reached to increase targets to 32.5 % for efficiency and to 32 % for renewable energy by 2030, while the recast of the Renewable Energy Directive awaits formal approval [58]. As Eurostat reported for 2016, 25.4 % of the final energy in Europe was used in the residential sector. In Belgium, about 85 % of the final energy used in households was for space and water heating [64]. As such, in the residential sector, large potential for CO2 reduction lies in lowering energy use by improving energy efficiency. This requires retrofitting of the building envelopes in combination with more efficient and low-carbon systems, such as heat pumps, solar heating, combined heat and power (CHP), biomass boilers and district heating. Among these technologies, heat pumps are of interest to this thesis because of their larger expected impact on the electricity grid compared to CHP [158], but also because of their increasing adoption in residential buildings. Heat pumps are seeing steady market growth the last years in Europe, and the total stock, about 10.6 million systems in 2017, is expected to double twice until 2030, according to the European Heat Pump Association [60]. Compared to this business-as-usual scenario, higher deployment could be achieved if renovations are increased [227], as a consequence of the energy efficiency and renewable targets the European Commission put forward. In fact, currently (2018) all three regions in Belgium provide subsidies for installation of heat pumps, particularly for renovations [26, 54, 67]. At the same time, distributed PV installations are steadily increasing in Belgium, after a surge between 2008 and 2011 responding to government incentives [96]. Currently subsidies for PV are coming to an end in Belgium, nevertheless, 85 % of new residential buildings choose to install a PV system in order to comply with the energy performance regulations in Flanders [228]. In 2017, almost 10 % of households owned a PV system in the country [96]. This thesis will further focus on rooftop PV, as most common in residential applications, compared to building-integrated systems.

Heat pumps and PV: Potential and challenges

2.2

|

9

Heat pumps and PV: Potential and challenges

PV systems employ solar cells to generate electric power from sunlight, taking advantage of the photovoltaic effect. During the operation of PV systems, no carbon emissions are produced, and solar energy is admittedly renewable. As such, PV is considered a renewable and low-carbon technology, which can contribute to achieving the targets of CO2 emission reduction. While large solar installations may generally be more efficient, distributed PV could reduce grid losses, if well balanced with local demand [171]. In Belgium, small installations, below 10 kW, represented more than half of the total installed capacity in 2017 [96]. Heat pumps extract energy from a heat source, either air, water or the ground, and transfer it to a heat sink, for instance the water in a hydronic heating system, by using auxiliary energy, most often electricity. The Renewable Energy Directive considers heat pumps with sufficiently larger output compared to the primary energy used as renewable energy technologies. The CO2 emissions attributed to heat pump operation can be very low or even zero, if the electricity used comes from renewable sources, which is why heat pumps are also seen as low-carbon technologies. This means, however, that the potential of heat pumps for CO2 emission reduction significantly depends on the electricity mix [17, 170, 209]. Since higher renewable shares are foreseen in order to achieve climate change mitigation, heat pumps are most often included in pathways towards decarbonization of the heating sector [35, 50, 147]. Current improvements in efficiency and communication technology, and the increasing renewable electricity generation further promote the adoption of this technology [66]. Both heat pumps and PV have a crucial role to play in the energy system decarbonization. However, there are also challenges related to their deployment, from socioeconomic aspects to environmental concerns and technical limitations. The initial investment cost and payback time may reduce the attractiveness of both heat pumps and PV for households [171, 209]. Policies, regulations and incentives can play an important role to this respect [208]. While both technologies have been described as low-carbon, there are environmental impacts that are often not taken into account, which emanate from the systems’ life cycle. In particular, many refrigerants used today in heat pumps can have severe effects on the ozone layer, and even contribute to global warming. European policies aim to phase down the use of refrigerants with high ozone-depleting and global warming potential [62]. Considering the production, installation, operation and decommissioning of heat pumps, the prospects may be less attractive [77]. Similarly, indirect emission can also be calculated for PV, which depend on the type of energy used to manufacture those systems [188]. While these will burden the emission records of the manufacturing country, it should not be forgotten that climate change has no geographical barriers.

10 |


The issues mentioned above are very important, however there are also technical challenges restricting the adoption of those technologies. For PV, the available surface area, orientation and potential obstacles may limit the profitability of those systems in some buildings, while solar irradiation in each location is also determinant. Wide deployment of heat pumps may be limited by the climate in a certain region, especially for air-source heat pumps, whose performance decreases at lower ambient temperature [209]. Ground-source heat pumps, on the other hand, have large installation costs, especially when little space is available, requiring vertical boreholes [208]. Furthermore, not all houses are suitable for heat pumps, since these work better for lower condenser temperatures, requiring larger emission surfaces and better insulated dwellings [170, 208]. The restrictions described above determine whether house owners will decide to install a heat pump or PV in their building. Assuming, however, that this decision is made, several challenges arise for the electricity system. PV generation is highly variable, creating a problem in balancing demand and production. For large shares of intermittent renewable generation from PV and wind, storage or transmission expansion could be necessary [27]. Furthermore, for high penetration levels, the electricity demand of heat pumps, and the injected power from PV could have significant impacts on the electricity system. At the system level, vast deployment of heat pumps could increase the peak demand and ramp rate, requiring additional generation and transmission capacity, as well as more flexible power plants [133, 182]. The focus of this thesis lies on the last type of technical issues that are reviewed here, namely local grid constraints. Since residential applications of heat pump and PV are connected to the LV distribution grid, large impacts can be expected at that level. Many studies reviewed the potential impacts of distributed PV on the LV distribution grid [81, 112, 137, 168, 171]. The major technical issues are voltage level variations, in particular overvoltage, unbalance, reverse power flow, islanding, increased power losses, frequency variations, harmonic distortion and transient voltage variation. Furthermore, voltage variation would result in frequent operation of voltage regulation devices, while reverse flow can affect protection equipment. Various voltage control strategies have been proposed in literature, including generation curtailment, grid reinforcement, batteries, voltage regulation devices such as on-load tap changers, and other more sophisticated emerging methods [29, 81, 137]. The interaction of PV systems with the grid needs to be taken into account when assessing PV integration scenarios, because of potential generation curtailment or additional costs for specialized equipment, storage or grid reinforcement. Heat pumps can significantly increase the electricity demand in LV grids, also resulting in technical issues, in networks that were not designed to accommodate such loads. Increased penetration levels can lead to overloading of the network

Review of literature on the assessment of LCTs

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components, such as the cables and transformers [157]. Large current can cause overheating and potentially damage those components, or even result in blackouts. Furthermore, larger current leads to higher energy losses. Undervoltage and voltage unbalance can also result from large loads, such as heat pumps and electric vehicles [129]. Other power quality concerns include fast voltage magnitude variations, harmonic distortion and voltage transients [20], similar to the impacts of PV. In Section 3.2 more details will be given for some of the described issues. Same as for PV, the large deployment of heat pumps should be considered together with measures to alleviate the impact on the grid. Indeed, because of the presented potential problems in the electricity grid, demand side management (DSM) will be necessary in the future, utilizing smart grid technologies [66, 166]. Thanks to the naturally available thermal storage in building envelopes, heat pumps can offer an important amount of flexibility, which can be used for grid balancing and increasing renewable integration [66, 73, 170, 191]. However, in order for this smart grid to materialize, infrastructure for smart metering and control need to be established, and all stakeholders to accept and promote a dynamic market [238]. Despite the potential of DSM, it is still a challenge to investigate it in combination with detailed district-level energy simulations [6], because predictive controls are rather complex and computationally expensive [66]. For this reason, the potential of DSM is also not incorporated in this thesis. As such, this work provides an estimation of heat pump and PV grid impacts assuming no interventions. The results can be subsequently used to inform studies investigating DSM or other solutions.

2.3


This section first gives examples of studies and applications where grid constraints are generally not taken into account, when LCTs as heat pumps and PV are considered in large numbers. As literature suggests, integrated approaches considering all energy vectors are needed, however computational limitations need to be first overcome. Section 2.3.2 then presents an overview of studies that specifically focus on quantifying the grid-related impacts of LCTs. Last, Section 2.3.3 makes a synthesis of the findings from this literature review.

2.3.1

The need to consider grid interactions

Energy saving or CO2 mitigation potential of heat pumps and PV at the scale of a country or region is often assessed using bottom-up approaches. These may

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use archetype buildings and systems, or housing stock models based on a large number of example buildings [10, 11, 72, 78]. In most cases, detailed models of buildings and systems are used, however the grid is typically assumed to acts as an infinite and lossless electricity storage system for each building [10]. As such, reported benefits may be overstated, since technical constraints and potential reinforcement costs are not considered. Pudjianto et al. [181] argued that massive grid reinforcements would be needed in the UK distribution grid in order to reach the 2050 decarbonization targets with heat pumps and electric vehicles. Furthermore, PV generation curtailment could be significant if voltage rise is not controlled in the local grid, not yielding the anticipated production [15]. As already mentioned, heat pumps could be controlled to improve self-consumption in buildings or balance renewable generation at the system level. As Patteeuw et al. [169] report, models that assess demand response potential of heat pumps and other flexible building loads may take either a demand side or supply side perspective, or use an integrated approach. The latter takes into consideration technical aspects of both the demand and supply. However, in most cases, the supply side representation does not include distribution limitations. For instance, [170] assumed an arbitrary cost for potential grid reinforcement in their calculation of CO2 -abatement cost for replacing gas boilers with heat pumps in Belgium. Few studies, nevertheless, include medium-voltage (MV) distribution grid simulation, calculating losses and voltage profiles in their assessment of demand response [237]. At district level, the view that an integrated modeling approach is required has gained popularity in the research community. For instance, the Annex 60 of the International Energy Agency “Energy in Buildings and Communities Program”[94] that ended in 2017 and has its continuation in the International Building Performance Simulation Association “Project 1” [93], had as objective to develop computational tools that allow buildings and energy grids to be designed and operated as integrated systems. The Annex60 Modelica library resulted from this effort. Nevertheless, in their review of district energy system modeling, Allegrini et al. [6] show that tools combining detailed building and electricity network simulation are relatively few. They furthermore found that power quality concerns are not adequately considered, and that the incorporation of demand-side controls is still challenging [6]. The need for multi-energy approaches in urban planning was also advocated by Page et al. [165]. They proposed a simulation and optimization environment for urban systems, that takes into account losses and potential reinforcement requirements in thermal or electrical networks, using a clustering approach to represent networks. Similarly, McKenna et al. [146] recently presented a method to generate optimal system configurations and operation of buildings in neighborhood clusters, considering the necessary network upgrade costs. Morvaj et al. [150] developed a framework to optimize the design and operation of buildings and district energy systems, taking into account electrical grid constraints. They have,


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13

for instance, optimized the systems of a small test district, including heat pumps, PV, micro-CHP and district heating network, with the objective to minimize carbon emissions and costs [151]. While such tools can be very useful for specific cases, the current run time would prohibit the use on multiple grids and scenarios, despite simplifications in load flow calculations [151]. Overall, grid constraints to the deployment of LCTs start to be considered in district level analyses, albeit with high computational cost or simplifications. Largescale studies, at regional or national level, consistently overlook issues related to the local distribution grid. However, LV grid impacts may be important for high penetration of LCTs, as suggested by Baetens et al. [15], Navarro-Espinosa and Ochoa [158], Pudjianto et al. [181] and other studies presented in the next section. As such, additional effort to take them into consideration in the problems mentioned here could be justified. This thesis will explore ways to do so inexpensively, as discussed further in Section 2.4.

2.3.2

Grid impact assessment

Investigation and quantification of the impacts LCTs have on LV networks have been the focus of an increasing number of studies, as those technologies become more common. To get an overview of the approaches that are used, this section presents the relevant literature, first on the integration of PV, which is often assessed independently of other LCTs, and then on heat pumps or combinations of LCTs.

Distributed PV systems Distributed electricity generation, and in particular PV systems, have been the focus of several studies with respect to their integration in LV grids. Some papers focus on specific problems and solutions related to PV, examined only for test networks and demand conditions. For example, Moshövel et al. [152] studied the potential of battery energy storage for reducing the grid impact. Mokhtari et al. [148] proposed a DC link to improve generation-load balancing and avoid curtailment in residential applications. In [125], a method for voltage regulation of distribution grids is presented, based on reactive power control and coordination through telecommunication of multiple PV inverters. Manito et al. [141] analyzed transformer aging related to increasing PV penetration rates. More general analyses of grid impacts were also performed on a limited number of representative, existing or test networks, often used as case studies to demonstrate the proposed methodology. For example, Refs. [75, 215, 236] have investigated the impact of increasing PV penetration on voltage and load indicators in example

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Belgian, UK and Swedish networks respectively. Gonzalez et al. [76] furthermore used the same Belgian networks to demonstrate a method for estimating critical penetration levels, with regard to the voltage margin available to distribution system operators (DSOs) for operating off-load tap changers. Harmonic distortion effects were evaluated for a real distribution grid with PV and electric vehicles by Tovilovic and Rajakovic [219]. Harmonic distortion from PV inverters was also studied by Du et al. [45], but focusing on measurements and modeling of the inverter. Several real feeders in the USA served to demonstrate the importance of using realistic PV generation profiles with high spatial resolution in grid impact studies [160]. Simple test networks were used to study the influence of PV modeling approach in impact analyses [52], to demonstrate a method for PV location and size optimization aiming to enhance voltage stability and reduce grid losses [56], and to test a methodology for probabilistic power flow with Latin hypercube sampling [111]. Recent literature has focused on developing probabilistic methods for grid impact assessments or network planning, using Monte Carlo approaches [117, 118, 155, 205]. However, due to the large simulation requirements, only one or two specific networks were analyzed. Watson et al. [231] carried out the laborious task of simulating an entire LV distribution system in New Zealand, with the purpose to analyze impacts of PV integration and explore voltage regulation measures. In the aforementioned papers, the most commonly assessed quantities were the voltage magnitude, voltage unbalance, cable and transformer loading, reverse power flow and grid losses. Many of the studies used three-phase unbalanced power flows with resolution below 15 min [52, 75, 117, 160, 215, 219, 231]. Conclusions vary depending on the examined networks and assumptions, showing problems intensify with increasing installed capacity, highly depending on the load localization on the grid. In these studies, grids are loaded only with present-day domestic electricity demand, which is represented as profiles resulting from stochastic models [155, 215, 236], smart meter data [75, 117, 118], or transformer level measurements [160, 205, 219]. The latter two approaches are not flexible enough to assess changes in buildings and their loads in the future.

Heat pumps and combinations of LCTs Regarding heat pumps, several studies have looked into their impacts on LV distribution networks using simulation. Earlier works assumed balanced grids or large time intervals, for instance Refs. [15, 139, 201]. However, since heat pumps, PV as well as electric vehicles may often be connected on a single phase, the resulting impacts may be underestimated if balanced conditions are assumed [129, 157]. Unbalanced three-phase load flow was included in [4], where the impact of heat pumps, PV and electric vehicles on an example grid was studied. Furthermore, Arnold [9] simulated an urban and a suburban LV German grid for future penetration


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scenarios for heat pumps, PV, CHP, thermal and electrical storage. Different voltage control measures were investigated to reduce the observed problems. While most studies employ steady-state analysis, focusing on component loading, voltage levels and losses, Akmal et al. [3] looked into the effect of heat pump start-up on transient voltages and Bottrell et al. [23] investigated the impact on harmonic power quality. Some studies have considered DSM in the context of LV distribution grids. Müller et al. [153] discussed a concept of DSM that accounts for grid current conditions, and proposed a platform to analyze control strategies in districts. Other papers also investigated the effect of heat pump control on grid impacts in LV grids [161, 181, 240]. Given that optimization requires much computation time, all these approaches often require simplified models or power flow analysis, and are implemented to few network cases. Also related to optimization, the work of McKenna et al. [146], and that of Morvaj et al. [151] were discussed in Section 2.3.1. These focused on optimizing the systems in districts, taking into account potential reinforcements. However, the large computation requirements limit their applicability to specific network cases [151]. The work of Baetens [12] and that of Navarro-Espinosa [156] are reported last, as they are the most relevant to this thesis. Navarro-Espinosa conducted an extensive probabilistic analysis of the impacts several LCTs have on real UK LV feeders [156]. It is the only work, to the author’s knowledge, that combines a probabilistic approach with a wide range of networks. The final analysis included a total of 128 feeders, derived from a GIS-database, and technologies such as heat pumps, PV, microCHP and electric vehicles. Furthermore, Monte Carlo iterations allowed to obtain probabilistic results for each feeder, LCT and penetration level. An analysis was also made to correlate several feeder characteristics with the maximum penetration of each LCT before problems occur. Based on those correlations, it was proposed to create lookup tables to estimate this maximum penetration level. These, however, take into account one parameter at a time, and would thus be of limited use. Metamodels, as presented in the next section, could provide an improvement to that respect. Furthermore, in [156] heat pump and CHP profiles were based on measured consumption of CHP in 16 houses. As a result, it is impossible to link building-related characteristics to the calculated impacts, thus prohibiting the evaluation of potential building retrofits. The present thesis, for this reason, takes a more building-oriented perspective on the issue. Last, the methodology of NavarroEspinosa [156] evaluates the impacts for only one day, and therefore cannot capture other potentially interesting quantities, such as the weekly evaluated voltage, the annual demand, backfeeding, or energy losses (see Section 3.2 for more details on those quantities). The work of Baetens [12] contributed to the development of the openIDEAS framework and Modelica IDEAS library [13, 16, 110], used in this thesis. It enables detailed building thermal simulation, stochastic occupant behavior modeling and

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unbalanced power flow simulation of LV grids. The same framework was used by Baetens to evaluate grid-related integration cost of heat pumps and PV in lowenergy dwellings, for two typical distribution networks, and few penetration and upgrade scenarios [12]. Important inputs for the present thesis are taken from that work, especially with regard to the network characteristics, as described in the next chapter.

2.3.3

State-of-the-art: Recommendations and limitations

Section 2.3.1 has shown that grid restrictions to the deployment of heat pumps and PV is rarely considered in large-scale studies. At the same time, district-level tools begin to incorporate local grid effects, but with large computational burden. To investigate ways for simpler grid impact assessment, the findings of the literature review in Section 2.3.2 are used to help determine an appropriate methodology. Most recent studies focusing primarily on the analysis of grid impacts use unbalanced three-phase load flow calculations, with at least quarter-hourly time step. Navarro-Espinosa and Mancarella [157] and Li and Crossley [129] demonstrated the impact of balanced loading assumptions on the estimated voltage and thermal problems. Moreover, Navarro-Espinosa [156] showed for the case of PV systems that voltage violations were underestimated with input resolutions below 10 min. Besides, the European Standard EN 50160 [53] prescribes voltage limitations for 10 min averages, as detailed in Section 3.2. The present thesis, therefore, adopts unbalanced three-phase load flow, combined with high resolution simulated loads. The requirement for a probabilistic approach is also highlighted in several of the reviewed papers [111, 117, 118, 129, 156, 205, 219]. These probabilistic studies, however, while taking into account the uncertainty in load and generation profiles, they only investigate a limited number of grid cases. As Arnold [9] points out, since every grid is different, it is important to assess grid impact for a variety of grid topologies and equipment. Therefore, this thesis proposes an approach to consider LV networks of different sizes and components. Common in a majority of available studies is the purely electrical point of view, which is often based on measured demand or simplified representation of building related loads. Beside potential loss of accuracy, this approach also prevents studying the benefits of building retrofitting on network relief, such as presented in [12]. The present work aims to emphasize the building perspective in grid impact studies. Last, despite the progress in computing power witnessed in recent years, the reviewed studies reveal that there are still limitations with regard to grid impact analyses, in particular for probabilistic approaches. For instance, many studies employ one or few typical days for the impact assessment [4, 52, 117, 129, 151, 158,

Metamodeling

| 17

160, 215]. Furthermore, recent research has focused on developing simplified LV network models [199], or exploring high performance computing for grid impact analyses [173]. Nevertheless, except for the computation time, probabilistic analyses on multiple networks require much data collection and modeling effort. Therefore, to offer a quicker and simpler evaluation, suitable for preliminary estimation of the impacts, Navarro-Espinosa and Ochoa [158] suggested the creation of lookup tables. This thesis proposes to generate metamodels instead, which also require little effort when used, but they allow to capture more complex relationships, offering better accuracy. Such models can also be incorporated into assessment tools, such as those described in Section 2.3.1. Details will be presented in the following section. The methodology for grid impact assessment proposed by this thesis should combine requirements that were identified as important or missing from the current state-of-the-art impact studies. These requirements, which will be incorporated in the framework presented in Chapter 3, are: d unbalanced three-phase load flow, d high input and simulation temporal resolution, d probabilistic assessment, d multiple different grid cases, d building-related parameters, and d a method to provide simple and quick assessment.

2.4

Metamodeling

In order to render the probabilistic framework more valuable for large-scale studies and other assessments of LCTs, methods are needed to overcome the inherent complexity of probabilistic grid impact assessments, in particular with respect to data and computation requirements. Metamodels offer this possibility, since their main purpose is to approximate results of the complex model, using few important inputs and requiring (much) less prediction time. This section explains the concept of metamodeling and presents applications in related topics. Based on this short review, the potential of metamodeling for grid impact analysis is discussed thereafter. Details on the metamodeling methodology proposed in this thesis will be given later in Chapter 5.

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2.4.1

What is metamodeling?

In many engineering and other problems, simulation models are built to represent the real systems, whose behavior we want to investigate. The purpose of these models is to replicate the system’s response, in order to assess different scenarios, perform sensitivity analysis or optimize a set of system parameters. However, as we strive to increase accuracy of the simulation model compared to the real system, the former often becomes computationally very demanding. As a result, the analysis or optimization becomes a very slow process. In order to accelerate the process, a metamodel can be used, also called surrogate model, emulator or response surface. A metamodel approximates the response obtained from the more complex simulation model, while being computationally cheaper to evaluate. It is constructed with a data-driven approach, which utilizes simulation results of the complex model, evaluated at few carefully selected points. The metamodel is generally not concerned with the processes occurring in either the real system or simulation model, but merely tries to fit the simulated response as close as possible. For this reason, such metamodels are often described as black-box models. The main steps of the metamodeling procedure include defining the experimental design, fitting the metamodel and evaluating its performance on test data. The procedure is explained in more detail in Chapter 5, where also the different types of metamodeling techniques relevant to this work are presented. An important distinction will be made in Section 5.3 between time-series metamodels and scalar response metamodels. This work will only focus on scalar metamodels, because time-series predictions for multiple different grid cases and uncertain inputs is very challenging, as explained in that section. Figure 2.1 gives a schematic representation of the relationships described above. The real system is modeled with a simulation model, based on domain knowledge and appropriate assumptions, which depend on the required accuracy. If the real response is available from experiments or measurements, the simulation model may be validated or even calibrated with those data. The simulation model is in turn approximated with a metamodel, that is fit and validated with the simulated response. The metamodel provides a fast approximation of the simulation model’s response, and by extension, an approximation of the real response. It can be then used for various analyses that require many simulation runs, such as design exploration and optimization, uncertainty and sensitivity analysis or for future predictions. Note that, sometimes metamodels can be built directly upon the observed real response, without intermediate simulation model, in a similar approach as machine learning and artificial intelligence. In this thesis, however, metamodels are used as presented in Figure 2.1, based on the probabilistic simulation results.

Metamodeling

Physical system

Real response

knowledge assumptions

validation?

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19

Simulation model

Simulated response

approximates approximates

fitting validation

approximates

Metamodel

Metamodel response

Design space exploration Optimization Uncertainty analysis Sensitivity analysis Prediction

Figure 2.1: Schematic representation of the relation between the real physical system, the simulation model and metamodel. To understand how metamodels are of interest to this work, more specific explanation of the used terms is given. The real system here denotes all LV distribution grids with heat pumps and PV that need to be evaluated. The response is the grid impact, expressed with a set of indicators, presented in Section 3.2. The detailed model represents the entire probabilistic framework that allows to obtain those indicators for all grid cases. This framework is described in Chapter 3, and it includes the definition of a design space with grid cases, as well as all simulations of buildings and the load flows. Finally, the metamodel is built upon the results of this framework, mapping grid cases to the estimated impacts. The potential use for such metamodels will be further discussed in Section 2.4.3, but first an overview is given of metamodeling applications in energy-related topics.

2.4.2

Review of metamodeling in energy-related applications

Metamodeling has been used in various engineering fields, and recently also for building performance assessment and energy use forecasting. Various studies have employed metamodels to compute energy demand and thermal comfort indicators for individual buildings, for instance references reviewed by Østergård et al. [163] and Kumar et al. [122]. Those were based either on measured responses or on building energy simulation results. Metamodeling techniques were also implemented for model calibration [140] and building design optimization [49, 223]. Energy use for building stocks has been predicted with metamodels based on physical representation of buildings [131], or historical data [127]. Few studies have looked into the spatial distribution of urban energy use, with relation to building and household properties [91, 232]. Time-series energy use metamodels were developed

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for individual buildings [7, 41, 122, 233, 239], building clusters or the entire power system [185, 239, 241]. Applications of metamodels can, furthermore, be found in electricity spot price forecasting [136] and voltage stability assessment of energy systems [194, 211]. Of all reviewed literature, only Nault et al. [154] have focused on neighborhood-level indicators that are not based on measured energy use or representative buildings. They generated metamodels for neighborhood solar performance assessment based on parametric modeling of neighborhood configurations. Their work evaluated the deterministic energy requirements and daylight potential, however, the approach is similar to the one adopted in this thesis. More specifically, the metamodels were created based on simulation of the neighborhood-level model for different neighborhood configurations. As such, parameterization of the neighborhood characteristics was first done, and the simulated cases were selected based on design of experiments. A similar approach is followed here, as detailed in Section 3.3, with the difference that each simulation is furthermore repeated many times to produce a probabilistic output. Of course, the quantities investigated and the simulation tools are entirely different as well. To the author’s knowledge, no applications of metamodeling to grid-related districtlevel indicators exists. Therefore, Chapter 5 develops a metamodeling methodology based on the review of publications more focused on mathematics and statistics.

2.4.3

Potential of metamodeling for grid impact assessment

It has been already stated that the purpose of metamodels in applications relevant to this work is to substitute complex models with simpler ones, that require fewer inputs and less prediction time. The metamodel can then be used to evaluate the response of interest in much more locations of the input space, allowing to make faster future predictions, explore different scenarios, optimize parameters, and perform uncertainty and sensitivity analysis. Because of these capabilities, metamodels are also attractive for grid impact assessment, since the detailed probabilistic framework is computationally expensive, and it also requires a lot of detailed inputs, as will become clear in Chapter 3. The elaborate simulation results can serve to understand better the phenomena leading to the observed grid impacts, so it is a necessary step in researching and recognizing the issues. From there on, a simpler and quicker prediction model is useful to support decision making in various levels. With regard to grid impact assessment, metamodels can be used to approximate a series of indicators pertaining to maximum component loading, extreme voltage levels, annual consumption and other quantities described in Section 3.2. The probability of

Conclusion

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exceeding certain limits can be furthermore assessed with a probabilistic framework. Except for indicators used in this thesis, many other variables could be computed based on the simulated time-series response, depending on the interest of different stakeholders. For instance, indicators could involve the frequency or duration of overloading, or the percentage of consumers affected by voltage problems. Economic assessments are not included in this thesis. Nevertheless, based on the predicted grid impacts, costs could be calculated in a subsequent step. For example, grid reinforcement costs could be estimated based on the probability of voltage violation or overloading for different grid designs. Metamodels that can give a preliminary estimation of grid impacts, in terms of indicators described above, can be useful in several applications. Policies concerning the adoption of LCTs like heat pumps and PV could choose to focus more on specific types of neighborhoods and networks, those that are expected to host more LCTs without problems. Furthermore, such policies on country or regional level could be founded on more realistic estimates of the technical limitations and potential grid upgrade requirements. Knowledge of the distribution network current state would be required for such task, however. Therefore, depending on the availability of those data, relatively small scale applications may benefit more from LV grid impact metamodels. In urban planning, for instance, the need to integrate different district energy systems is increasing as new technologies emerge, however the multi-scale and multi-sector nature of urban planning poses a great challenge [28]. For early stage planning, metamodels for grid impact evaluation could give an estimate of the expected problems and direct the decisions related to the adoption of LCTs and network planning. Finally, a tool based on metamodels could help DSOs design and operate their distribution networks by foreseeing future impacts without need for detailed simulations.

2.5

Conclusion

This chapter has elaborated on the broader context this work is situated in, addressing the issue of climate change and the efforts made in the European Union to mitigate it. These efforts are concentrated on reducing carbon emissions, by improving energy efficiency and integrating more renewable energy sources. In the residential sector, heat pumps and PV systems are the most commonly adopted installations that aim to improve building energy performance, alongside with thermal insulation. Both heat pumps and PV have a prominent position in energy transition pathways, however their wide deployment is expected to face technical issues, in particular with regard to their integration in the LV distribution grid. Overloading and voltage

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problems could be severe, requiring grid upgrades and load management. Therefore, better understanding and quantification of the potential issues is essential to allow decarbonization of the energy system. This thesis specifically aims to analyze the LV grid impact of heat pumps and PV. In order to determine a methodology for grid impact assessment of LCTs, a review of the relevant literature was conducted. This study identified aspects that were highlighted as important in the reviewed works, as well as aspects missing from the current literature. More specifically, unbalanced three-phase load flow calculations with high resolution inputs was presented as indispensable by many studies. Furthermore, increasing interest in probabilistic assessment approaches could be observed, which tries to estimate grid impacts considering uncertainties in the loads. Both requirements are, therefore, taken into account in the methodology developed in this thesis. It was also found that most literature assumes buildings as a given boundary condition, therefore not allowing to examine implications of different building designs or renovation scenarios. This work provides deeper understanding of the influence buildings have on the resulting grid impacts. Additionally, an important limitation of current literature is that only specific real or example networks are analyzed. Therefore, the methodology developed here encompasses a broader range of network cases, in an attempt to provide more generalizable assessments. Disadvantages of detailed and probabilistic grid impact assessment that have become evident from the review are the complexity and computational requirements of such approaches, which often lead to simplifications and restrictions of the scope. For this reason, this thesis proposes to develop metamodels in order to provide a simpler and faster tool for grid impact assessment, that can be easily used by different stakeholders. This chapter has presented the main concept of metamodeling, also providing a small summary of metamodeling applications in energy-related topics. Since no similar applications in grid impact assessment were found, literature in mathematics and statistics will be analyzed in Chapter 5 in order to determine an appropriate methodology to generate metamodels for grid impact indicators. These metamodels could serve as decision support instruments in policy assessment, as well as in urban planning and network design and management.

3

Framework for probabilistic grid impact analysis

The previous chapter has discussed the potential impacts of heat pumps and PV on the electricity distribution grid. Furthermore, it identified the limitations of current literature in grid impact assessment of low-carbon technologies (LCTs). This chapter, therefore, develops a framework that enables grid impact analysis for different networks, taking into account the influence of uncertain building properties in a probabilistic way. The aim of this framework is twofold: First, it aims to analyze the impact of LCTs on residential low-voltage (LV) grids, concentrating on the importance of demand diversity in terms of building characteristics and load coincidence. The framework furthermore performs this analysis on a broad range of grid cases and LCT penetration scenarios, in order to provide an informative assessment. Chapter 4 presents the results of this analysis. Second, the framework aims to provide a basis for further metamodeling of important grid impact indicators, which is necessary in order to make these results accessible for different stakeholders. The general metamodeling methodology is discussed in Chapter 5, while application of specific techniques is presented in Chapter 6. The following sections describe the general approach and scope of this probabilistic framework, the quantities of interest, the various types of input factors and the experimental design. Next, the simulation models and their specific implementation are presented. The last section investigates the influence of certain modeling assumptions and methodological choices.

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3.1

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General methodology and assumptions

General methodology Figure 3.1 gives an overview of the steps followed to develop the framework for grid impact analysis, leading to the creation of metamodels (presented in Chapter 5). First, the quantities of interest and specific grid indicators are defined, which determine the requirements for simulation models, as well as the types of metamodels that should be used in a later stage. Section 3.2 presents those grid impact indicators. Then, the various inputs to the framework are specified, defining the experimental design. The proposed framework evaluates grid impact quantities for a variety of different LV distribution islands in a probabilistic way, that is, taking into account uncertainties related to the loads. Therefore, distinction should be made between grid parameters and uncertain inputs, as described in Section 3.3. The different grid cases are first constructed with a main experimental design that tries to efficiently represent the possible combinations of network characteristics (grid parameters). A metamodel fitted to the results, allows to afterwards interpolate to other similar cases within the limits of the design. For each grid case, the Monte Carlo Impact Indicators

Building Stock

Simulation

Quantities of interest for grid impact evaluation. Loading and voltage indicators at feeder and island level.

A set of electricity demand and generation profiles, derived from detailed building simulations.

Annual electrical network simulations using models from the IDEAS Modelica library.

Input Design

Grids

Metamodel

Experimental design including the main array defining grid cases, and replications to take into account uncertainties.

A set of low-voltage distribution islands with various configurations and penetration levels for low-carbon technologies.

Mathematical approximation based on simulation results. Simpler and faster prediction of grid indicators for highlevel analysis.

Figure 3.1: Main steps of the methodology for probabilistic impact analysis and metamodeling.

General methodology and assumptions

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25

method is adopted to reproduce the variability in grid impact indicators resulting from various uncertain inputs. This approach is commonly used for uncertainty propagation [124], where the output distribution is approximated by evaluation of the model for many input combinations, called replications. Section 3.3 discusses the probabilistic framework inputs in a general way, establishing the link with metamodel features and the necessary experimental design. Specific details on the simulation inputs are given in Section 3.4, where the models and assumptions are extensively described. Simulation models are generated following a two-step approach, as presented in Section 3.4. To represent uncertainties in the loads, a building stock is first created, which comprises generation and demand profiles. The latter are a result of detailed building energy simulations. Variability is taken into account by sampling geometric and thermal properties from predefined probability distributions, as well as by using a stochastic model for occupant behavior. In a second step, LV grid cases are specified. In this work, grids are defined at the level of LV distribution islands, comprising several three-phase radial feeders, all connected to the same mediumto-low voltage (MV/LV) transformer (see Figure 1.1 on page 2). One feeder is the main focus of each grid case (from now on referred to as feeder case), with the rest of the island represented in a simplified way, as described in Section 3.4.2. Networks of varying characteristics and several LCT penetration levels are included. For each feeder case and LCT penetration level, loads are sampled from the building stock in several replications using a Monte Carlo approach, to take into account uncertainties in the loads. All replications of the feeder cases are then simulated for one year, providing the necessary material for the analysis and metamodeling presented respectively in Chapter 4 and Chapter 6.

General assumptions For the proposed probabilistic framework, computational effort increases fast as more grid cases are investigated. The number of grid cases increases when more grid parameters are varied and more options are included. Therefore, certain assumptions are made to maintain a tractable problem. In Chapter 1, the general scope of this work was already limited to Belgian residential LV grids, with single-family dwellings. Furthermore, air-source heat pumps and rooftop photovoltaic systems (PVs) were chosen as LCTs, since they represent the most common low-carbon solutions in buildings that interact with the electricity grid. For both technologies, one set of technical characteristics are used for all buildings, so as to limit the number of possible combinations. These are described in Section 3.4.1. Additionally, the adoption of these technologies is generally assumed

26 |


to be independent of occupants and building characteristics. Nevertheless, for heat pumps, a pre-selection of buildings that can potentially adopt a heat pump is made, as explained in Section 3.4.1. Afterwards, the allocation of heat pumps and PV to the buildings in a grid can be performed randomly and independently, allowing the option for houses to have both at the same time. Furthermore, as mentioned in Section 1.2, a reference case for the networks is considered, without measures taken to mitigate the grid impacts from these LCTs. Therefore, the networks are assumed without special equipment for voltage control, and smart control, based on grid-related or other signals, is not in place. Moreover, the operation of heat pumps is assumed not to be affected by voltage drops in the network. This allows for decoupling building and grid simulations, since demand is not influenced by network conditions, drastically reducing simulation time. Last, this framework assumes that buildings and occupants in a given neighborhood do not interact with each other, not in terms of heat transfer or shadow casting, nor in terms of occupant interaction. As such, buildings and their occupants can be modeled independently of other buildings. It is also assumed that no correlation exists between household types. This means that neighborhoods with a majority of elderly couples, for instance, are not considered.

3.2

Quantities of interest

As discussed in Chapters 1 and 2, residential LCTs, such as heat pumps and PV, are expected to have important impacts on the LV distribution grid. The increased electricity consumption and generation, combined with higher peak simultaneity may result in congestion and power quality issues. This section describes quantities related to the evaluation of LCTs integration in LV grids, elaborating on the pertinence of each. Furthermore, specific grid impact indicators are deduced from those quantities, which are further used for analysis and metamodeling in this work. Table 3.1 on page 30 provides a detailed description of those indicators.

Load characteristics Residential LCTs aim to reduce carbon emissions, either by replacing CO2 -producing equivalent equipment, or by supplying CO2 -free energy. In both cases, electricity is often involved, such as for heating with heat pumps, and distributed generation with rooftop PV. As a result, traditional loads in the local distribution grid of a residential neighborhood change significantly. A first level evaluation needs to look into the transformation of load patterns, in terms of annual energy fluxes, peak demand,


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27

the shape and simultaneity of loads. This analysis may be performed by simple aggregation of loads, without need for grid simulation. It is assumed that these loads are not subsequently altered by any grid-related control strategy. In particular, taking the feeder as reference unit, annual electricity generation E g and consumption by the households E d are of interest. Based on feeder-level load profiles, the net electricity consumption E d,net can be calculated, expressing the electricity demanded from the utility, having taken into account electricity produced and consumed within the feeder. The latter allows to assess the feeder-level selfconsumption, similar as used for individual buildings [226], providing an indication of the impact on the medium-voltage (MV) grid. In this value, energy losses in the distribution system are not included. The feeder-level load shape may be graphically analyzed with the help of loadduration curves. It is interesting to examine the peak demand P d , peak generation P g , total peak or disaggregated by end-use, for different penetration levels of the LCTs. These peak values will depend on the simultaneity of various loads in the feeder. Since distribution grid design and sizing has been traditionally based on assumed simultaneity or diversity factors k s , an update on those values for scenarios with increased penetration of LCTs is necessary.

Exchange with backbone grid Conventional electricity distribution systems were designed for unidirectional power flow, from central power plants to the consumers. As explained above, future grids will need to satisfy higher demand and accommodate decentralized electricity generators, which under certain circumstances may create reverse power flows and inject power to the MV grid. Therefore, evaluation of the energy exchange at the distribution transformer is necessary in order to quantify the influence of a given distribution island on the overall system. Quantities of interest include the annual net electricity consumption E t,d,net and backfeeding E t,bf , as well as the related peak net demand P t,d,net and backfeeding P t,bf (see Table 3.1). These quantities are calculated based on power flow at the distribution transformer, and therefore include losses within the distribution island (depending on the model). They can be used to quantify the necessary generation and distribution capacity of the overall system, as well as determine whether special protection needs to be foreseen for backfeeding.

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Grid component overloading Grid impact is most often expressed as overloading of network components, mainly the distribution transformer and feeder cables. Overloading occurs when the thermal rating of the component is surpassed. Excessive current flowing in the component results in large heat dissipation and temperature rise. Excessive temperature then causes damage to the protective insulation of transformers and conductors. In transformers, temperature rise is due to both iron and copper losses, which respectively depend on the voltage and current flowing through windings. As these are independent of load power factor, transformers are rated based on the apparent (total) power S n . Distribution transformers may occasionally be loaded above nameplate rating, however, sustained overloading can reduce the normal life expectancy of a transformer’s insulation, normally assumed at 180 000 h [97]. The loss-of-life for common Mineral-Oil-Immersed transformers is calculated based on the winding hottest-spot temperature [97]. The models proposed in Section 3.4 allow calculation of this quantity [12], but with significant increase in computation time. Therefore, this option was not used in this work. Instead, the maximum apparent power S max served to evaluate transformer loading, as a fraction of the nameplate rating S n . For cases where unity power factors are assumed for the loads, the real power P could be used as well. Similar to transformers, cable insulation deteriorates at high temperatures. It is common in grid impact studies to evaluate the maximum utilization rate of the feeder cables, expressed as the maximum root mean square (RMS) current I max over the cables’ rated capacity [158]. To define indicators for overloading, an averaging time period needs to be defined. As overloading is a thermal issue, this period should be close to the thermal time constant of the component. However, as Bollen and Rönnberg [20] point out, literature reports varying time constants from minutes to hours. Grid impact studies have used averaging intervals between 10 min and 1 h [19, 157]. Here 30-min averaging intervals are taken for overloading assessment of both feeder conductors and transformers, as a median solution, also because IEEE Std C57.91 [97] refers to periods less than 30 min as short-term loading. The impact of averaging period is, nevertheless, explored in Section 3.5.2. Significant overloading implies that the component should be replaced with one of higher rating, in order to avoid failure with important consequences. While strict overloading limits don’t exist, studies on LCT integration often set the nominal capacity as the limit for component replacement [12, 146, 157].


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Voltage fluctuations: Overvoltage and undervoltage Several power quality issues may arise from the introduction of renewable technologies in future grids [20]. Grid impact studies based on simulation commonly evaluate voltage quality in terms of slow voltage fluctuations, in the time scale of 10 min [118, 146, 151, 153, 158]. The European Standard EN 50160 [53] requires the 10-min RMS voltage to remain within ±10 % of the nominal (Un = 230 V ) for 95 % of time each week, and between +10 % and −15 % Un for all time. Therefore, in this work, 10-min averaged voltage indicators are created at the feeder level. Specifically, the analysis looks into the yearly minimum 10-min averaged line-to-neutral RMS voltage at the farthest connection of the feeder (Umin ). Furthermore, the weekly 5th percentile of 10-min voltages is calculated for a rolling window of one week, starting 95%w every day. The yearly minimum of all 359 values is taken as indicator (Umin ) to be compared with the minimum weekly limit of 90 % Un . Last, the maximum value over the entire feeder (Umax ) is used for overvoltage assessment. Chapter 5 provides further explanation on how variations of these indicators may be more useful, such as whether violation of the limits occurred or not.

Other power quality issues Other phenomena affecting power quality in distribution systems include voltage unbalance, fast voltage magnitude variations, and harmonics [20]. Voltage unbalance occurs in a three-phase system when voltage magnitude and angle are different among the three phases. This often results from uneven distribution of large single-phase loads, such as heat pumps and PV. Voltage unbalance has negative effects on three-phase connected equipment, such as induction motors [103]. Some grid impact studies have also investigated voltage unbalance in networks with single-phase PV integration [75, 192, 205] and heat pumps [12], showing important unbalance may occur. In this work, however, this phenomenon was not studied. In the time scale below 1 min, voltage magnitude variations may also cause problems, such as flicker and tripping of sensitive loads, for example, adjustable speed drives. Voltage transients, with duration of few milliseconds, may occur from lightning or the start-up and switch-off of heavy loads. Last, voltage and current harmonics in the grid degrade power quality, increase thermal losses and reduce the life of equipment. These are caused by non-linear loads, for example, computers, battery chargers and variable-speed drives, such as used by a modulating heat pump. All these phenomena need to be captured at very small time scales, and are commonly only studied based on measurements and experimental results [3, 85]. Consequently, they are out of scope for the present work.

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Table 3.1: Overview of indicators used in this thesis for grid impact analysis in Chapter 3 and metamodeling in Chapter 6. Symbol Description Load characteristics (feeder level* ) Ed Annual electricity consumption by households in feeder (MWh) Eg Feeder annual electricity generation (MWh) E d,net Feeder annual electricity consumption (net demand) (MWh) Pd Feeder peak electricity demand, 30-min averaged (kW) Pg Feeder peak electricity generation, 30-min averaged (kW) Exchange with backbone grid (transformer level* ) E t,bf Distribution island annual electricity backfeeding (MWh) E t,d,net Distribution island annual electricity consumption (net-demand) (MWh) P t,bf Distribution island peak electricity backfeeding, 30-min averaged (kW) P t,d,net Distribution island peak electricity net demand, 30-min averaged (kW) Pt Distribution island signed peak load, 30-min averaged (kW) Grid component overloading S max Transformer maximum apparent power, 30-min averaged (kVA or pu† ) I max Feeder cable maximum current, 30-min averaged (A or pu) Voltage quality (feeder level) Umin Minimum feeder 10-min averaged RMS phase voltage (V or pu) 95%w Minimum 5th weekly percentile of 10-min averaged RMS phase voltage (V or pu) Umin Umax Maximum feeder 10-min averaged RMS phase voltage (V or pu) * Transformer-level indicators are subscripted with a “t”, while feeder-level indicators have

no subscript, for simplicity.

† pu (per-unit): Load as a fraction of the component rated capacity. Voltage as fraction of

the nominal voltage Un =230 V.

3.3

Inputs and experimental design

The probabilistic framework evaluates grid impact indicators in a probabilistic way, by including uncertainties using the Monte Carlo approach. This section first discusses the various factors influencing the quantities of interest described in Section 3.2. These factors are then classified into different types, which helps selecting an appropriate experimental design that serves for both probabilistic analysis and metamodel building. The discussion here remains more generic, with specific input values and distributions given in Section 3.4, where the models and assumptions are described. The last part of this section elaborates on the experimental design, a key element of the adopted methodology.


3.3.1

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Factors influencing low-voltage grid impact indicators

In order to decide which factors will be included in the probabilistic analysis, expert judgment and domain knowledge should ideally be combined with sensitivity analysis. As Saltelli et al. [197] point out, sensitivity analysis is essential for models used in impact assessment for policy support, and it is, therefore, prescribed in national and international guidelines. Global sensitivity analysis methods are preferred for non-linear problems, because they explore better the input space and consider interactions [196, 216]. These methods include, among others: screening, regression-based methods, variance-based methods, and metamodelbased methods [101, 196, 216]. In this work, the main inputs used to generate the building stock and grids were selected based on domain knowledge and available literature, as explained in the following. Sensitivity analysis based on partial correlation coefficients is performed for different grid parameters in Section 4.2. We have seen in Section 3.2 that LV grid impact analysis is concerned with quantities describing load characteristics, such as the peak and total demand and generation, as well as component loading and voltage quality. Influential factors for these grid impact indicators are related to the district electricity demand and production, and to the network properties and topology. Among other generation technologies, such as wind turbines and micro-CHP, rooftop PV systems are more common for electricity generation in Belgian dwellings. Electricity produced with this system depends mostly on the employed technology, and the available solar radiation. It is assumed that PV systems in all buildings have the same technical characteristics, and they are all located close enough to be subjected to almost the same solar radiation. This assumption is common in similar studies [12, 75, 156, 215]. Furthermore, since the neighborhood configuration is not explicitly modeled, except for the network, local shading is not taken into account. As a result, the total production in the neighborhood depends on the weather, the number of PV units, and the capacity, orientation and tilt of each unit. District electricity consumption depends on several factors: physical, technical and socio-economic [135]. Since the scope of this work was limited to heat pumps as consuming LCT, electricity demand may be split in two parts: the base-load electricity demand for household appliances and lighting, and the heat pump electricity use. The former depends on the installed lighting power, building design and sunlight, on the appliances, their power requirements and, most importantly, their use, which is directly influenced by occupant behavior and choices [235]. For this framework, variation in the base-load is handled by the stochastic occupant behavior model used to generate electricity demand profiles (Section 3.4.1). The impact of weather on the lighting load is considered, but not the building geometry. A

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pool of demand profiles is generated, corresponding to different households, which are then randomly assigned to buildings. As such, no additional input is specified to describe the base-load demand. Heat pump electricity consumption depends on the heating system itself, including technical characteristics and various settings, on the outdoor temperature, as well as on heating requirements. There is a vast number of parameters that define the heating system performance, from controls to the emission system and production unit. Within a probabilistic methodology, it is impossible to vary these systems and their parameters for each building individually. Therefore, the same heating system is implemented for all buildings that are selected to have a heat pump. In this thesis, only a different power factor (ratio of real to apparent power) has been investigated, while the rest were kept fixed for all buildings and simulated feeder cases. Variations in other parameters or control methods could be assessed in a similar way. The heating requirements include domestic hot water and space heating needs. Hot water requirements and space heating set-points are both linked to occupant preferences, and are therefore handled in the stochastic occupant models. Space heating needs furthermore depend on the entire heat balance of the building’s heated zones, which is a function of weather conditions and building characteristics. Much literature on building energy performance assessment has identified parameters influencing space heating requirements the most, although with focus on annual energy needs. For a temperate climate like in Belgium, apart from the weather and occupant-related inputs, the following building properties are commonly considered in literature [88, 100, 120, 164, 223]: building size (surface area or volume), compactness, envelope insulation level, glazing properties, air change rate, window-to-wall ratio, and orientation. The same parameters define the nominal capacity of the heat pump, since the system is sized based on the design heat load [159]. The uncertainty in these parameters is taken into account while generating a building stock dataset, which samples values from the probability distributions assigned to these properties (see Section 3.4.1). The above have discussed factors influencing the production and consumption of electricity for individual dwellings. Grid loading depends on the number of buildings, PVs and heat pumps, on the individual load profiles and the simultaneity of those. In this thesis, the randomness characterizing the simultaneity of demand profiles is reproduced by using a stochastic model for household base-load demand profiles and heating schedules, and by randomly sampling buildings and households to populate the grids. Voltage and current in the different parts of the network moreover depend on technical specifications of the network, such as the transformer and conductors, the reference voltage at the MV/LV transformer, on the location of each load on the grid, and the distance between nodes [12, 75, 157]. It is important that different configurations are considered, for instance with varying average cable length between consumers. The locations of loads must be also varied.


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Based on all aspects described here, the next section defines the different inputs for the probabilistic framework. Section 3.4 then provides all the details regarding modeling choices and specific values assigned to those inputs.

3.3.2

Input for probabilistic framework and metamodel

In order to develop an appropriate experimental design for the probabilistic analysis and metamodeling, it is important to first distinguish between the different types of inputs, and how these should be treated in the framework. Here, the framework aims to assess the probabilistic grid impacts of increasing LCT penetration, for a variety of different feeder cases. The metamodels built upon those results should help estimate for a given feeder the allowable LCT penetration levels, or compare solutions to mitigate the technical issues, for instance. Therefore, the main inputs relate to the definition of those feeder cases, while other inputs represent the variability, or uncertainty, within each case. The following three types of inputs will be described: grid parameters, uncertain factors, and scenarios. Table 3.2 gives an overview of the selected inputs in this thesis, indicating their categorization into the mentioned types. More details can be found in Section 3.4.

Grid parameters Grid parameters, as the name suggests, define the different grid or feeder cases to be evaluated. These are parameters that must be easily determined for an existing grid, and they are expected to have important contribution to the resulting impact indicators, justifying the creation of a separate feeder case for analysis. In probabilistic design terminology, these would be called decision, control or controllable factors, or even design parameters [42, 214, 221]. Indeed, some grid parameters are in fact controllable and optimizable, under certain circumstances. These are the network technical characteristics, such as the cable types Ca, the transformer rated capacity S n and tap changer settings (Uref ), and the general buildings’ construction quality Q. Changing these parameters in the model represents potential grid reinforcements or building renovations applied as solutions to mitigate voltage or overloading problems in a grid. Similarly, the LCT penetration levels (r HP , r PV ) may be assumed variable, when studying a network’s hosting capacity. However, there are other grid parameters which are in practice fixed for a given grid. These are, for example, the number of buildings N and the distance between building connections l avg , or qualitative parameters, such as the neighborhood type T (urban or rural). The latter is used to define the range of other grid parameters (l avg , NI ) or uncertain factors (see Section 3.4). Because of these fixed parameters, the terms controllable or design are not used in this context to avoid confusion. Table 3.2 gives a summary of the chosen grid parameters in

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Table 3.2: Inputs for the probabilistic framework, classified as grid parameters, uncertain factors or scenarios. The specific values used are given in Table 3.5. Grid parameters T Neighborhood type N Buildings in feeder l avg Average cable length between buildings r PV PV penetration level in feeder r HP Heat pump penetration level in feeder Q Construction quality Ca Cable strength Sn Transformer rated capacity Uref Reference transformer voltage NI Buildings in rest of island r PV,I PV penetration level in rest of island r HP,I Heat pump penetration level in rest of island cos φ Heat pump power factor Weather Specific buildings in feeder or island Occupants in each building Location of heat pump systems on feeder Location of PV systems on feeder

X X X X X X X X X X X X (X)

Uncertain factors

(X)* (X) (X) (X) (X ) (X)

Scenarios

X X

X X X X

* Potential other options, not adopted in this thesis.

this work, while details are provided in Section 3.4.2. Grid parameters determine the feeder cases to be investigated, therefore, they constitute factors of the main experimental design discussed in the following subsection. For the same reason, they are used as potential features in the metamodels, as elaborated in Section 5.4.

Uncertain factors Uncertain factors represent the second type of inputs, also called noise or environmental factors [214]. These represent the variability in results for each feeder case, which stems from uncertainty and randomness in the loads and their distribution on the grid. They include stochastic effects, such as human behavior, and other inputs which are uncontrollable or difficult to observe in reality. For example, the properties of each building in the grid, the location of LCTs, etc. It may be possible to observe or measure some of them for an existing specific network, such that they are no longer uncertain—set aside potential measurement uncertainty. Nevertheless, the probabilistic grid impact assessment methodology aims to provide assessment for a variety of feeders, for which such detailed information is generally unavailable. Furthermore, it is difficult to express inputs such as the location of an LCT or the


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occupancy data with a single parameter. Therefore, these uncertainties cannot be represented in the experimental design by a set of parameters with predefined probabilities, nor can they be used as metamodel features. Instead, a building stock with varying properties and a pool of stochastic occupant profiles are generated. Replications are then used for each feeder case, every time resampling buildings and occupants, and assigning them to random locations on the grid. More details are given in Sections 3.4.1 and 3.4.2. Note that uncertain inputs may as well include specific parameters with their probability distributions. In fact, some of the grid parameters mentioned above could be considered as uncertain inputs, under some circumstances. For instance, if a probabilistic assessment is required for a certain feeder, regardless of the conditions in the remainder distribution island, then the number of buildings and LCTs in the island could be seen as uncertain. Also, the reference transformer voltage could be seen as an uncertainty related to the voltage variations in the MV grid, instead of a transformer setting (see Section 3.4.2 for more details). The next subsection gives some information on how to treat such uncertain inputs in the experimental design.

Scenarios Scenarios describe inherently uncertain situations affecting the analysis, for which we may desire a separate evaluation. An example is weather conditions. Comparison between feeder cases may yield different results if another climate or weather file is used. Technological scenarios can also be analyzed, where, for instance, PV systems have better efficiency, or heat pumps have different performance. The latter is included in this work, where the power factor determining reactive power consumption of heat pumps is changed, assuming a common value for all heat pumps. This scenario is investigated, since the power factor has been shown to introduce significant variation in grid impact indicators [157, 176]. Note that, if a different power factor was assigned to each individual heat pump, this parameter could also be considered as an uncertain input. However, in this work a separate evaluation is desired. Specifically, the weather and power factor scenarios are analyzed in dedicated sections, 4.7 and 4.8 respectively, while for the main analysis and metamodels only one option is used. Especially when scenarios relate to the use of different time-series inputs, such as the weather data, comparison of results is easier when all simulations are repeated for this new scenario. In this case, a separate metamodel may be built for each scenario. Alternatively, for metamodels where the scenarios can be represented by a single parameter, it could be beneficial to include the scenario parameter in the main design, to furthermore limit the number of simulations.

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3.3.3


Experimental design

The experimental design determines the levels (values) of all factors (inputs) for each simulation run needed for the probabilistic analysis and/or for metamodel building. Figure 3.2 shows the experimental design used for the probabilistic simulation and the corresponding dataset created for the metamodel. For this work, the grid parameters determining the feeder cases constitute the main design array, with replications used to represent the uncertainty due to random effects. Random sampling of buildings, occupants and their allocation on the grid are repeated for each replication of every feeder case (input combination or design point).

Replications for uncertainties To include uncertainties, Taguchi’s approach for robust design suggests the use of crossed array designs, consisting in the cross-product of an inner array with control factors (here the feeder cases) and an outer array with noise factors (uncertain inputs) [214]. In this way, all control factor combinations are subject to the same uncertainties, allowing better comparison of design options [221]. However, other researchers have proposed the use of a combined array, where control and noise factors are both columns of the same array, reducing the necessary simulation runs [204, 234]. Here, neither the crossed nor the combined array approach is possible. A combined array would require that the noise or uncertain factors are represented by a set of parameters, which is not the case. Potential uncertain parameters not used here could be included in this way, however. Furthermore, it is not possible to maintain the same randomly sampled buildings, occupants and LCTs for all design points, because of the varying feeder sizes, neighborhood types and LCT penetration levels. Therefore, in the proposed probabilistic framework, uncertainties are included via replication. When replications are used to obtain a probability distribution at each design point, the metamodel is often built for a specific statistic of this distribution, for instance the mean (Figure 3.2), as explained in Section 5.3. While some rules of thumb exist to determine the number of replications for metamodeling the mean, more research is still needed [113]. Here, 20 replications are used for the main analysis, and the impact of this choice is investigated in Section 3.5.3 and Section 6.4.3 for metamodeling.


1 ··· m

37

Replications (uncertain inputs): random allocation of buildings, occupants & technologies

Grid (and scenario ) parameters g1 ··· gk

y

1

x1

···

xq

1 ...

...

zi

i

zi

i

n

...

zi

...

Feeder cases / design points

|

n Framework design space

Simulation output z

Metamodel Metamodel output features

Figure 3.2: Overview of the simulation experimental design and the resulting dataset for metamodeling. The former produces a distribution of the output by employing replications. The metamodel uses as output a specific statistic of that distribution, for instance the mean. Main design for feeder cases To generate the design points of the main design, traditional Monte Carlo uses random sampling. However, nowadays other space-filling designs are preferred, which cover the design space more efficiently. As elaborated in Section 5.5, these include Quasi-Monte Carlo and variations of Latin hypercube sampling [70]. Latin hypercube designs (LHDs) and Quasi-Monte Carlo sampling are both prominent in literature, but the former are generally one-shot designs, not allowing sequential addition of points to improve convergence [105]. In this work, Sobol’ sampling is chosen to generate the feeder cases with the Matlab functions sobolset and scramble. This design was created much faster than optimized LHD with the software available to the author, and was therefore preferred, since literature suggests they both perform similarly [33]. The experimental design has k = 12 factors, which are the grid parameters from Table 3.2, on page 34. Furthermore, the design size n was chosen as 1296, a multiple of the possible combinations of discrete input values (Table 3.5 on page 53), to ensure approximately equal representation of all cases. This design size is quite large, much larger than the rule of thumb n = 10 × k suggested in literature for an initial design size [132]. Section 3.5.4 investigates whether smaller sample sizes would be sufficient to obtain

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the same overall distribution of results, while Section 6.4.4 examines how metamodel performance decreases as this sample size is reduced. As these sections show, different conclusions may be drawn for different indicators and different uses.

Input correlations It is possible to furthermore introduce correlations between factors in the experimental design, if this correlation can be estimated, as further explained in Section 5.5. Here, no additional correlation was introduced, except for that resulting from the input definitions themselves. More specifically, the range of possible values for the average distance between buildings and the number of buildings in the island depend on the neighborhood type (see Table 3.5). This dependence relates to the network topology and classification criteria for neighborhood types, which should not differ significantly in the near future. Furthermore, the neighborhood type (rural or urban) and construction quality influence the building types (detached, semidetached, terraced) and range of buildings sampled each time. The building stock creation, as presented in Section 3.4.1, incorporates correlations between the various construction properties as well, also through their definitions. Potential correlations not considered include the selection of families in the buildings of each feeder, which are assumed here to be independent. Other correlations may exist between inputs when looking at the Belgian current or future situation in general. For example, smaller islands would tend to have smaller transformers. Therefore, if the purpose is to extrapolate findings to a region or country, the existing correlations may be needed to recreate feeder cases that are representative for this region, and that can be linked to specific frequencies in the total stock. However, when it comes to controllable grid properties, such as the transformer, cables, LCT penetration levels, etc., the probabilistic framework and metamodel aim to investigate all possible options. There is also no evidence, to the author’s knowledge, suggesting certain options are generally implemented together. Furthermore, strong collinearity should be avoided in regression analysis, as it may result in inaccurate coefficient estimates of regression models [149].

Treatment of scenarios In this thesis, scenarios are analyzed separately, with repetition of the entire experimental design as is, by only changing the scenario input. This allows for direct comparison of the scenarios, but also demands many simulations. As Figure 3.2 suggests, another option is to include scenario parameters in the main design array. This can be more efficient and useful if the scenario must be included in the same metamodel. However, it requires that a parameter represents the scenario. This is

Simulation models and procedures

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39

possible for the power factor scenario, but not as easy for the weather scenario. In the latter case, a categorical variable could be used to indicate the scenario number, or better, a more descriptive parameter, such as the heating degree days, maximum temperature, or other. This option, as already mentioned, was not investigated in the present work.

3.4


As mentioned in Section 3.1, variability in building properties is taken into account using a building stock, from which buildings are sampled to populate the feeders. As illustrated in Figure 3.3, creation of the load profiles constituting the building stock involves various simulation models, including a building thermal model, an occupant behavior model, and a PV model. The next section describes in detail those models and the inputs they use. Section 3.4.2 discusses the grid cases, describing the varied parameters and simulation models. For all building, PV and grid models, the IDEAS Modelica library was used, in the Dymola simulation environment. Modelica is a non-proprietary, object-oriented language that is appropriate for modeling multi-domain complex physical systems, providing flexibility for continuous development. The IDEAS library was specifically developed at the KU Leuven to integrate dynamic simulation of both thermal and electrical systems at district level, also providing models for stochastic occupant behavior [13, 16, 110]. Furthermore, it is publicly available at https://github. com/open-ideas. The IDEAS library has been already used to assess strategies for EV integration in buildings [224] and to estimate externalized costs from heat pump and PV integration in low-energy neighborhoods [12]. Among other applications, IDEAS has also been used for optimal control and design of buildings and building systems [110, 172] and district energy demand simulations [38]. Modelica, as a component-based modeling language, offers the possibility to incorporate any types of LCTs, storage and controls, using custom made or existing components of IDEAS and compatible libraries. While this tool can provide the detail needed for the analysis of peak loads and voltage, computational limitations may arise from such complexity, a burden especially noticeable in a probabilistic approach. In this work, simplifications are made and restriction of the simulated options to limit the simulation time requirements. Nevertheless, more advanced methods could be used to improve simulation speed of complex Modelica models [109]. All loads have been simulated for a period of one year, for the typical moderate climate of Uccle, Belgium. Hourly synthetic weather data for thermal simulations and minutely data for irradiation were obtained from Meteonorm v7.1.6 [190]. A

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Heating system

Envelope properties

BUILDING MODEL Detailed 2-zone model with heat-pump, radiators, and varying envelope thermal properties.

OCCUPANCY MODEL Space heating set-points, internal heat gains, hot water requirements and base-load demand.

PV MODEL 34° inclination, orientation West, East or South. Nominal power 230 W for one panel.

HEATING ELECTRICAL LOAD BASE-LOAD PV GENERATION

BUILDING STOCK

Figure 3.3: Creation of a building stock, consisting of different load profiles. Building, occupancy and PV models are used for this purpose. weather year with extreme cold winter and sunny summer was selected for the main analysis and metamodeling, which represents a 10-year extreme expectation for grid impact indicators. Nevertheless, Section 4.7 investigates the impact of different weather year selections. Temporal resolution for the load and grid simulations is analyzed in Section 3.5.2. As a result, 1-min simulations were carried out for the loads. The resulting profiles were subsequently averaged at 5-min resolution, to match the simulation step of grid simulations. Finally, the entire workflow for simulation preand post-processing are semi-automated by means of Matlab code.

3.4.1

Building stock

The building stock is created to represent variability in demand and generation, caused by differences between consumers, in terms of building characteristics, LCT adoption and occupants. To achieve this, a set of load and generation profiles are constructed, as illustrated in Figure 3.3, which are then sampled to populate the grids. To serve the aim and scope of this analysis, the building stock comprises typical Belgian single-family dwellings, optionally equipped with a PV system and an air-


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source heat pump for space heating and domestic hot water (DHW). The assumption is made that significant electric loads in households only include the potential heat pump and PV, and a base-load stochastic household demand. The latter includes receptacle loads for household appliances and lighting, as defined in the Stochastic occupant behavior section on page 47. In dwellings without a heat pump, heating is assumed to be provided by a gas boiler or other system not requiring electricity. Furthermore, potential large loads for special uses are not considered in this work. As a result, buildings without heat pumps are represented in the analysis solely by the base-load profiles and optional PV. Therefore, detailed building models are only needed for those dwellings that are equipped with a heat pump. While PV installation is generally only limited by the building’s geometry and orientation, a certain efficiency level is required to decide the adoption of a heat pump for space heating. In particular, it should be feasible to successfully deliver the required heat with low-temperature radiators of reasonable size, or floor heating. Therefore, buildings that install a heat pump are assumed to have undertaken at least light renovations, resulting in medium to small energy losses from building envelope and ventilation. Allowing a large range of thermal insulation quality, as described in the following, only the option of low-temperature radiators is investigated in the modeled buildings. The next section presents the assumptions and models used to generate the building stock load profiles.

Building parameterization As a first step to the building stock creation, buildings are parameterized based on their geometric and thermal properties, for which probability distributions are specified. The building cases are created using an optimized maximin LHD [92]. From the original design of size 100, two additional designs were generated using random column permutations, as performed by Janssen [105], in order to create 300 different building cases. The sample size 100 was chosen larger than the rule of thumb n = 10k of Loeppky et al. [132], where k is the number of parameters, here six, but small enough to limit the time required to produce the optimized LHD design. The building stock size of 300 was selected so that buildings can be sampled without replacement in the grids, and at the same time keeping computation and memory requirements low. Ideally, an analysis should be made to verify the convergence of important indicators depending on the building stock size. Due to time restrictions, this task was not performed in this thesis, as many sets of simulations would be required. Each building case of the building stock is translated to the equivalent simulation models, based on assumptions and simplifications detailed in the following sections. As explained later, three variants are modeled per building case, to represent detached, semi-detached and terraced dwellings.

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Table 3.3: Overview of the varied building input parameters and their considered distributions. Symbol

Description

Distribution

A

Floor area, m2

wwr ori Uop

Window-to-wall ratio, Orientation, U -value opaque, W/m2 K U -value floor†, W/m2 K Window type (Table 3.4)

n ach P 0,PV

Air change rate, h-1 PV rated capacity, kW

Terraced & Semi-detached: N (150,40)*, Detached: N (200,50) U (0.2, 0.5) South, West, East U (0.1, 0.6) ¡ ¢ min 0.1, U (Uop − 0.15,Uop + 0.15) • Air 4/15/4 (U -value=2.0), if Uop > 0.35 • Ar 4/15/4 (U -value=1.6) • Kr 4/15/4 (U -value=1.2) • Kr 4/8/4/8/4 (U -value=0.8) U (0.3, 0.8) U (3, 5), max: roof dependent

* N (µ, σ) denotes a normal distribution with mean µ and standard deviation σ.

U (a, b) denotes a uniform distribution between a and b. For discrete uniform distributions, the possible options are given. † Value referring to the steady-state U -value of a slab on ground.

Parameters were chosen, which have been shown to have significant impact on the space heating demand of dwellings [88, 100, 120, 164, 223]. Being able to investigate the influence of building characteristics on grid impact indicators is also an essential objective of this work. However, a previous related study showed that detailed parameters, such as U -values of individual building components, could not be directly linked to those indicators [180]. Furthermore, such detailed information is rarely available for real-life studies. Therefore, fewer and lumped parameters describing the overall building performance have been selected, as summarized in Table 3.3. These include three geometric parameters, a general thermal transmittance parameter, an air change parameter and a PV-related parameter. More information is given in the respective sections below.

Building structure Parametrization of the building geometry is based on four parameters: building type, floor surface area (A), window-to-wall ratio (wwr) and orientation (ori). The building type defines contiguity with neighboring houses and includes three options: detached, semi-detached and terraced. This parameter influences the definition of boundary walls, repartition of window area and floor plan dimensions ratio, as illustrated in Figure 3.4. Furthermore, it defines the probability distribution


ℎ

Night-zone ℎ/ 2

ℎ

Day-zone ori w

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43

If Semi-detached or Terraced: adiabatic wall ℎ =3m Detached: d = w Semi-detached or Terraced: d = 2w

d

Figure 3.4: Geometric model for the buildings. The depicted two-dimensional surfaces refer to the mean surface at the insulation layer. assigned to the floor surface area parameter, as seen in Table 3.3. It must be noted that the building type parameter is not sampled; instead, all three variants are modeled, for each combination of other parameters. This is done to ensure unbiased sampling of buildings properties in a feeder regardless of building type, since the latter depends on the feeder (neighborhood) type (Section 3.4.2). For instance, rural feeders contain a majority of detached dwellings, while urban feeders a majority of terraced dwellings. Floor surface area and window-to-wall ratio distributions were arbitrarily defined based on examination of related investigations and surveys [5, 36, 72, 106]. The generic geometric model shown in Figure 3.4 is used to calculate all surface areas, based on the sampled geometric parameters. Walls towards neighboring houses in case of semi-detached or terraced houses are modeled as adiabatic walls, assuming similar conditions on the other side. The building structure model comprises two thermal zones: the day-zone, representing living area and kitchen on the ground floor, and the night-zone including, for instance, bedrooms and corridors on the first floor. The two zones are connected with a common floor, but no air circulation. The air in each zone is assumed to be perfectly mixed, with uniform temperature. Internal heat gains are distributed arbitrarily as 70 % to the day-zone and 30 % to the night-zone. It is expected that more appliances would be located in those rooms, and that occupants would also spend more of their active time in kitchen and living room. Regarding envelope construction details, in order to obtain a realistic overall parameter, while also taking advantage of the detailed simulation model, the next procedure is followed. The average U -value of the opaque elements, Uop , is sampled. This parameter does not include windows, because their U -values are typically much higher than the average envelope U -value. Furthermore, window U -values can only

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Table 3.4: Window parameters. For all types, the frame-to-window ratio is 0.25. Type Air 4/15/4 Ar 4/15/4 Kr 4/15/4 Kr 4/8/4/8/4

Window U -value W/m2 K 2.0 1.6 1.2 0.8

Glazing U -value W/m2 K 1.43 1.10 0.75 0.61

Frame U -value W/m2 K 3.71 3.10 2.55 1.37

Glazing g -value 0.605 0.609 0.613 0.402

assume discrete values corresponding to different glazing types in the model. The distribution range for Uop was chosen to cover components of buildings built from 1991 [36] up to passive-house standards [99]. Window components are subsequently randomly sampled from the available options detailed in Table 3.4; simple double pane glazing is only an option for buildings with Uop >0.35 W/m2 K. Moreover, the floor U -value is randomly sampled within a range of 0.15 W/m2 K around Uop , and with a lower limit of 0.1 W/m2 K. The corresponding U -value of the remaining opaque elements, namely walls and roof, is then calculated to reach the sampled Uop . Using this approach extreme differences between building components resulting from random sampling of all individual U -values are avoided. Envelope components are designed as typical Belgian constructions, as proposed by the TABULA project [36], but with insulation layers of varying thickness, based on the desired component U -value. Thermal bridges are not taken into account by the model. An area-weighted average U -value of the entire building envelope Uavg based on all components is further used in the analysis. To prevent overheating in the summer, which is important in particular for heavily insulated dwellings, exterior shading devices are foreseen. These block 76 % of shortwave solar radiation reaching the window pane, and they are controlled based on both the indoor operative temperature (hysteresis between 22 and 25 °C) and solar irradiance (hysteresis between 150 and 250 W/m2 ). Active cooling is not considered, as it is uncommon in residential buildings in Belgium.

Ventilation system The total air change rate n ach is also a sampled parameter for the buildings, which accounts for both hygienic ventilation and infiltration. It was assumed that all houses with heat pump are also equipped with a balanced mechanical ventilation system with heat recovery, and are, therefore, relatively airtight at an average infiltration air change rate of 1.5 air changes (h-1 ) at 50 Pa pressure differential, or approximately 0.075 natural air changes. The remainder of n ach is allocated to the ventilation system, whose heat recovery efficiency was assumed at an average of 70 %. Both infiltration rate and recovery efficiency don’t satisfy the suggested limits for passive


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houses, namely 0.6 h-1 at 50 Pa and 75 % respectively [99], as they represent an average for all simulated cases. Furthermore, to prevent overheating and improve thermal comfort, bypass of the ventilation heat recovery is implemented when average daily outdoor air temperature exceeds 15 °C, approximately between May 15 and September 15 in Belgium.

Heating system As stated earlier, all modeled buildings in this work are equipped with an airsource heat pump, which is connected to the grid via a single-phase connection. The modulating heat pump is individually sized, and provides hot water to lowtemperature radiators in each thermal zone, and to a 200 L DHW storage tank. It was assumed that no buffer storage tank is used for space heating. A central room thermostat is located in the day-zone, and a thermostatic radiator valve in the nightzone. When heating of the stratified DHW tank is required, a 3-way-valve switches priority to the DHW circuit. The heat pump model is based on interpolation in a performance map retrieved from manufacturer data, which defines the heating power and electricity use as function of the condenser outlet temperature, the evaporator temperature and the modulation rate. Minimum modulation rate is set to 20 %. The performance map does not take into account (de)frosting of the evaporator. Furthermore, only active power use data are available, therefore the power factor is assumed to be unity. Based on the manufacturer’s data, the coefficient of performance (COP) of the heat pump is 3.17 at 2/35 °C test conditions and 2.44 at 2/45 °C test conditions for full load operation. The original nominal output is 7177 W at 2/35 °C, which is used to rescale results to the required nominal power. The latter is set equal to the design heat load Q des calculated as described by the European Standard EN 12831 [159], using a reheating factor of 16 W/m2 . Radiators are sized for the design heat load of each zone, and have a fraction of radiant heat output of 35 % and exponent 1.3. Furthermore, the nominal supply temperature for space heating Tsup is defined based on the average U -value of the building (Uavg ) as: Tsup =45 °C for Uavg < 0.35 W/m2 K, Tsup =50 °C for 0.35 ≤ Uavg ≤ 0.55 and Tsup =55 °C if Uavg >0.55. The nominal temperature difference between supply and return for the heat pump is 10 °C in all cases. The heat pump supply temperature set-point for space heating is based on a heating curve. More specifically, the heating curve requires the nominal Tsup for the design outdoor temperature of -8 °C and design room temperature of 21 °C, and has an overall minimum at 30 °C. The heat pump’s set-point is 2 °C higher than this curve, to account for potential thermal losses of the distribution circuit, not within the heated zones. Control of the heat pump is based on the measured indoor air temperature in the day-zone, and the set-temperature required by the occupants,

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with a 1 °C hysteresis on the room thermostat. Last, a 3 kW back-up instantaneous electric heater is activated to assist space heating in very cold days. These backup devices are installed by many manufacturers, despite them causing unwanted peaks and elevated electricity consumption. Their size in commercial heat pumps ranges roughly from 1 to 6 kW for single-phase connections depending on the heat pump capacity. To avoid frequent activation, usually a lock-out outdoor temperature limit is applied, above which the back-up is set out of operation. In this work, a conservative lock-out temperature of 0 °C was applied, below which the electrical instantaneous heater provides the necessary additional power to meet the required water temperature. The 200 L DHW tank is modeled as a stratified one-dimensional multi-node storage tank, as described in [37]. It is heated every night by the heat pump to a comfort level of 55 °C to cover most daily needs. The heat pump switches to DHW heating once the tank temperature, measured at the upper part, drops more than 4 °C below this setpoint. This regime takes advantage of potential favorable night tariff, while avoiding prolonged interruption of space heating during the day when it’s most needed. In the event of extreme demand during the day, the tank is reheated to maintain the minimum level of 43 °C. The demanded water temperature is assumed at 42 °C, to account for potential not modeled losses in the building’s hydraulic network. Additional power is provided by a 3 kW immersion back-up electric element when the tank temperature drops further below 38 °C. The water heating schedule differs for the 300 simulated building cases, gradually starting between 21:30 and 00:30 with a 5 h duration. In this way, diversity between consumers is taken into account, even though all are assumed to follow a certain advantageous tariff. Last, anti-legionella cycles are scheduled once a week during the evening, one hour after the daily heating starts [21]. The electrical immersion heater then boosts the water temperature from 55 °C to 65 °C. The day of the week varies from house to house. Last, it is assumed that the heat pump operation is not influenced by grid conditions. This is a necessary assumption in order to allow for decoupling of building loads and grid simulations. In practice, if voltage levels would become very low, it could be expected that heat pump operation would be interrupted, shifting the demand to later times. However, no specific information was found related to the behavior of heat pumps under these conditions.

PV system The PV installed rated capacity P 0,PV is also treated as a sampled parameter of the house, although it is only used in the grid simulations in the next step. The reason is that the sampled rated capacity is truncated if the available roof area is found insufficient. It is assumed that no more than 80 % of the roof area on one side can


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be covered by PV panels. For the comparison, each 230 Wp panel is considered to cover a surface of 1.4 m2 . The probability distribution range for P 0,PV is chosen between 3 and 5 kW, the latter being the maximum allowed capacity for a singlephase connection in Belgium [212]. PV generation is simulated based on the 5-parameter model of De Soto [39], as implemented by Baetens et al. [15], with properties taken for a 230 Wp commercial panel of 16.6 % efficiency. This model generates 1-min resolution PV production profiles for the required orientations, at an assumed optimal inclination for Belgium of 34 degrees. These are adjusted for the sampled capacity of each house. It is assumed that all houses are in a relatively small area where irradiation and sky coverage are identical for all. The electricity produced is first used to cover the building’s demand, and the remainder is injected to the grid. Furthermore, to avoid potential excessive feeder voltages, the PV system inverter can be disconnected when voltage reaches a predefined limit, for a minimum duration of 5 min. This limit is set to 10 % increase of the nominal feeder voltage, or 253 V, according to national regulations [212]. Note that, the in-home electrical circuits are not explicitly modeled; instead all loads are applied directly at the building’s switchboard, which is connected to the feeder with a single-phase connection. Additionally, the PV is assumed to only deliver real power, as is common for residential applications.

Stochastic occupant behavior Occupant behavior is modeled with the Python StROBe package of IDEAS [13]. Three hundred different households were generated stochastically with this program, which were then semi-randomly assigned to the buildings. The number of different profiles was chosen equal to the number of different building cases. Profiles were generated until the ratio of single-person households was close to national statistics, which was on average 35 % in 2018 [210]. While the trend is generally increasing, it can be also expected that in single-family dwellings the percentage would be lower than the general average, which includes apartments. It is also possible that the ratio varies between neighborhoods. As such, more data on household composition at high spatial resolution could help define ways to better represent variation. In this thesis, a small analysis is performed in Section 4.9 to evaluate the impact of different single-person household ratios on the simulated results. Each generated household contains annual profiles for the following: 10-min data for air temperature space heating set-points for day- and night-zone; 1-min internal heat gains produced by occupants, lighting and appliances; 1-min hot water requirements at 42 °C; and 1-min electricity demand. The latter, also referred to as base-load in this thesis, includes electricity demand for lighting, large and small domestic appliances and electronics. Only active power is included in this model.

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A set-back temperature of 16 °C was used in the temperature set-point profiles, in order to avoid excessive temperature drops during the night. To assign households to the buildings, the former are ordered based on total annual internal heat gains, and the latter based on floor area. Then, for each building, a household is randomly chosen from a 150-household range around the equivalent position in the ordered households dataset. For example, for the 200th building case, a household is randomly sampled between the 125th and 275th. Close to the extremities, the available range is truncated appropriately. With this technique, occupancy remains random but extreme cases are avoided, for instance very small houses with extremely large internal gains. The sampling procedure furthermore makes sure that all 300 buildings get a different household. All three variants of each building case, namely the detached, semi-detached and terraced house variant, are assigned the same household, since only one variant is used at a time.

Building simulation results Figure 3.5 displays simulation results for an example house, representing a moderately insulated large detached dwelling during the coldest week of the year. Space heating is controlled based on the air temperature, but the operative temperature, calculated as the mean of internal air and radiant temperatures, is also displayed as a better measure of thermal comfort. Due to the building’s large thermal mass, the operative temperature rises much slower than the air temperature, and remains below the set-point throughout the day. This building also cools down fast, as a result of high thermal losses. Additionally, the night-zone air temperature falls much below the desired levels during the evening, since the thermostat is situated in the day-zone, which then doesn’t require heating. These problems often occur in buildings, despite the use of auxiliary heating, due to inadequate control. For better comfort, often settings with higher set-back temperature would be preferred, as is generally also advised for heat pumps. In the bottom panel of Figure 3.5, we can see that the added load from back-up electric heaters is significant on cold days, not only leading to higher cost for the owner, but also straining the grid. Therefore, their use should be reevaluated in the future, and accompanied by more advanced control. A simple early heat pump start-up could still satisfy comfort while reducing electricity consumption and peaks in new and more energy efficient dwellings. The bottom plot in Figure 3.6 shows such a building would achieve comfort with a delay of only a couple of hours without auxiliary heating. This is not necessarily true for moderately insulated buildings. In the same very cold day, the set-point temperature was only reached after a delay of four additional hours, compared to the situation with back-up heating (top plot of

Temperature (°C)


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49

26

10

24

8

22

6

20

4

18

2

16

0

14

-2

12

-4 Tair, day Top, day Tset, day

10 8

Tair, night Top, night Tset, night

-6 -8

Tamb

-10

6 6 6

7

8

Heat pump Heat pump BAK

5

9

10

11

12

13

12

13

PV

DHW BAK Base-load

Electrical load (kW)

4 3 2 1 0 -1

6

7

8

9

10

11

-2 -3 Days (midnight)

Figure 3.5: Example of moderately insulated detached house (Uavg = 0.60 W/m2 K) during the coldest week of the year. Top: air, operative and set temperatures for both zones (left axis) and ambient temperature Tamb (right axis). Bottom: electrical loads per use (BAK denotes back-up). Figure 3.6). For such cases, auxiliary heating by means of a gas boiler, for instance, would be preferred in terms of reducing peak loads.

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26

Tair with BAK

Temperature (°C)

24

Tair without BAK

Tset

8

22

6

20

4

18

2

16

0

14

-2

12

-4

10

-6

8

-8

Uavg=0.60 W/m2K

6 0

Temperature (°C)

10

Tamb

6

12

18

0

6

12

18

-10 0

26

10

24

8

22

6

20

4

18

2

16

0

14

-2

12

-4

10

-6

8

-8

Uavg=0.28 W/m2K

6 0

6

12

18

0

6

12

18

-10 0

Time of day (hours)

Figure 3.6: Day-zone air temperature of two example detached dwellings with and without back-up heating (BAK) for the coldest day. Top: moderately insulated dwelling with Uavg = 0.60 W/m2 K. Bottom: well insulated dwelling with Uavg = 0.28 W/m2 K.

3.4.2

Grids

Grids in this thesis are parameterized to represent Belgian LV distribution islands, with ranges of variation inspired from related literature [12, 75]. Belgian distribution feeders are typically three-phase, four-wire, wye systems, with nominal voltage of 230/400 V at 50 Hz. A distribution island comprises several feeders connected to the same MV/LV transformer, as depicted in Figure 1.1 on page 2. According to the work


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of Baetens [12], who combined geographical and network data to construct typical Flemish distribution islands, the majority of MV/LV voltage distribution transformers (90 % of studied cases) supply between 20 and 200 residential consumers, through 1 to 8 radial feeders. Most individual feeders contain between 5 and 40 consumers, with a maximum of 65. Furthermore, the statistical analysis showed two main neighborhood types could be distinguished based on building types and density, namely rural and urban ones [12]. In a similar study, Gonzalez et al. [75] identified four different types, namely rural, semi-urban, urban and city feeders. Apart from the difference in building types, they also accentuate the difference in average length between households in those feeder types, from 7.6 m in the city, to 44 m in rural feeders. For the definition of feeder cases in this thesis, findings of both above studies are combined, as explained below.

Grid parameterization As seen in Section 3.2, grid impact analysis is concerned with voltage and current indicators at the feeder level, as well as transformer loading at the distribution island level. Optimally, the entire distribution island should be analyzed in detail, in order to obtain the most accurate evaluation. However, detailed island simulation not only creates computational issues, but also increases dramatically the potential inputs to vary in the framework. On the other hand, simulation of isolated feeders prohibits the assessment of transformer loading, while also neglects the voltage drop at the transformer due to the remainder of the island [176]. Section 3.5.1 provides detailed analysis of different modeling approaches. As a result of this analysis, it is found sufficient to employ an intermediate approach, where the feeder of interest is modeled in detail, while the remainder distribution island is represented by an aggregated load. Figure 3.7 gives a schematic representation of this dummy island approach, where two feeder types are defined. This modeling approach was shown to produce relatively small deviations in grid impact indicators compared to full island simulations, while providing important reduction in CPU time (Section 3.5.1). The dummy island approach allows focusing on the details of a specific feeder, while also taking into account loads in the remaining island. Furthermore, compared to using few typical islands, such as in [12, 175], dummy islands allow for independence between feeder results and more variation, resulting in more efficient metamodeling. With this dummy island approach, parameterization of the grid is done at two levels, the main focus lying on the studied feeder. At the island level, the transformer rated capacity S n is a first grid parameter, as specified in Section 3.3. Three options are chosen, seen in Table 3.5, representing the most common sizes in Flemish grids [12]. Furthermore, the transformer secondary reference voltage (no-load voltage) Uref is considered as grid parameter, representing potential different setting of a tap changer transformer. While Uref may generally change in time, for instance with

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Rural (R)

MV grid

NI D l

MV grid

D l

8m D anywhere lex

SS

D

...

D

N

≈20% semi-detached (S) ≈4% terraced (T) Rest: detached (D)

Urban (U)

NI

3m

STTTTTTT

...

≈20% semi-detached (S) Rest: terraced (T)

TS N

lex

l l

lavg, N, NI: grid parameters

lex~

(lavg,

N·lavg 4

)

l=

N·lavg–lex N–1

Figure 3.7: Rural and urban feeders with dummy representation of the remaining distribution island (NI buildings). Grid configuration depends on the neighborhood type and l avg , with an additional random exceptional length l ex . on-load tap changers, three distinct constant levels were chosen, at the nominal voltage and ±5 %. Except for voltage regulation purposes, voltage deviations at the transformer may be also due to variations in the MV grid, not explicitly modeled here. In this context, Uref could be rather regarded as an uncertain input, and treated differently in the experimental design (Section 3.3). For the rest of the distribution island, except the feeder of interest, only three parameters are used, since no network is modeled. To specify the aggregate load, the number of buildings NI , and the percentage of houses with heat pumps r HP,I and PV systems r PV,I are needed. Grid parameters also include the cable type Ca, the number of buildings N , heat pump and PV penetration levels r HP and r PV , and the average distance between building connections l avg . Two cases are chosen for the cable types, namely a strong and moderate version, as specified in Table 3.5. It is assumed that any feeder replacements or new installations would use strong cables with section of 150 mm2 aluminum per phase [12]. Weak cables with sections smaller than 70 mm2 are not investigated, since they would probably be replaced already for small penetration levels of heat pumps. Previous studies have shown voltage or thermal problems would occur at small penetration levels [12, 174]. The number of buildings is chosen based on the findings in [12]. Heat pump and PV penetration levels are assumed from zero to 100 %, in order to have a complete assessment. While at the country level, these numbers would not be realistic, it is possible that specific neighborhoods reach 100 % penetration at some point in the future. l avg represents the total cable length divided by the number of buildings, and it is dependent on the neighborhood


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Table 3.5: Inputs for the probabilistic framework. Symbol Description Grid parameters T Neighborhood type Q Construction quality in feeder N Buildings in feeder r PV PV penetration level in feeder, % r HP Heat pump penetration level in feeder, % Ca Cable strength l avg Average cable length between buildings, m Sn Transformer rated capacity, kVA Uref Reference transformer voltage, pu ‡ NI § Buildings in rest of island r PV,I § PV penetration level in rest of island, % r HP,I § Heat pump penet. level in rest of island, % Scenarios cos φ Heat pump power factor, Weather Uncertain factors Specific buildings in feeder or island Occupants in each building Location of heat pump systems on feeder Location of PV systems on feeder

Values {rural,urban} {old, renovated, new} U {10, 45} * U (0,100) U (0,100) {moderate, strong} † R: U {5, 10}, U: U {15, 30} {160, 250, 400} {0.95, 1 , 1.05} R: U {0, 130 − N }, U: U {0, 200 − N } U (0,100) U (0,100) 1 Extreme year Random sample from building stock, based on Q Random sample from occupants pool [174] Random Random

* U {a, b} and U (a, b) denote a discrete and continuous, respectively, uniform distribution

between a and b.

† Strong feeders: Al 4×150 mm2 ; moderate feeders and less than 15 buildings: Al 4×70 mm2 ;

moderate feeders and more than 15 buildings: Al 4×120 mm2 .

‡ pu (per-unit): voltage as fraction of nominal voltage U =230 V. n § Equivalent for the entire distribution island: N = N +N , r t I HP,t = (r HP N +r HP,I NI )/Nt , and

r PV,t = (r PV N + r PV,I NI )/Nt .

type, an additional parameter. Distinction is made between rural and urban grids, as in the work of Baetens [12]. The neighborhood type influences the total number of buildings, the average cable length and building types, as shown in Table 3.5 and Figure 3.7. Rural feeders and islands are assumed to have about 20 % semi-detached dwellings placed in pairs, 4 % terraced, and a majority of detached dwellings. Urban ones have also about 20 % semi-detached dwellings, and the rest are terraced. Locations of the semi-detached are chosen at the ends of urban feeders, as would be in a real street, and randomly in pairs for the rest. Figure 3.7 additionally displays the difference in configurations and cable distances for the two types. A random exceptional length l ex is introduced in the beginning of urban feeders, or in a random

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location in rural feeders, to represent exceptional distances found in real feeders. One last quantitative grid parameter is used in the framework, namely the general construction quality Q for the feeder. This parameter defines the building stock subset from which buildings are sampled into the feeder. Its purpose is to better illustrate the influence of building properties on the grid. Three construction quality cases are created, to represent new, partially renovated and old neighborhoods. Buildings in a new neighborhood are assumed to be well insulated. Buildings in an old neighborhood are assumed to have undertaken at least a moderate renovation, justifying the adoption of a heat pump. These buildings are the least performing of the simulated stock, while a broader mix can be found in renovated neighborhoods. To account for construction quality when assigning buildings to the main feeder, sampling is performed, without replacement, from subsets of the dataset, ordered based on the average U -value (Uavg ). The arbitrarily chosen ranges shown in Figure 3.8 are used for sampling from the ordered building stock, depending on Q. Buildings for the remainder of the island are sampled from the entire building stock, without replacement, having removed those sampled in the main feeder. As explained before in Section 3.3, the experimental design generates 1296 combinations of the grid parameters of Table 3.5, the design points. Then, 20 replications are simulated for each design point, which aim to take into account uncertainties in uncertain factors, such as the location of each load on the grid. For each replication of every design point, different buildings with heat pumps and optional PV are sampled for the feeder and rest of the island, taking into account the construction quality. Every time, the feeder topology is also altered by changing l ex in Figure 3.7, and new locations in the feeder are randomly chosen for buildings, heat pumps and PV. When no heat pump is present, only the stochastic base-load electricity profile is assigned to that building. renovated

new 1

old

150 300 Buildings ordered from low to high Uavg

Figure 3.8: Building stock subsets used to sample buildings of a given construction quality Q . To define the subsets, buildings are ordered based on Uavg . Simulation model As already mentioned, grids are simulated independently of buildings, due to computation time restrictions. The pre-simulated load and generation profiles, averaged at 5-min intervals, are aggregated and then applied at the building-grid

Sensitivity to modeling assumptions

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connection point. Self-consumption is taken into account in that way. Dwellings are connected with a single-phase connection, alternating between the three phases. Since heat pumps ans PV are randomly allocated to the buildings, unbalanced conditions may occur. Year-long grid simulations are then executed in the Dymola software, with a 5-min output resolution. The IDEAS library provides models for three-phase unbalanced power flow analysis of LV grids, using a quasi-stationary method and assuming constant frequency [225]. Furthermore, the voltage at the transformer secondary is assumed to be constant throughout the simulation. The Uref parameter is used, as explained above, to take into account potential settings of a tap changer. Last, cable and transformer properties are defined in Tables 3.6 and 3.7.

Table 3.6: Cable properties used in simulation.

EAXVB 1 kV 4×70 mm2 EAXVB 1 kV 4×120 mm2 EAXVB 1 kV 4×150 mm2

Resistance Ω/Km

Reactance Ω/Km

Current rating (buried) A

0.461 0.269 0.206

0.072 0.071 0.070

225 280 315

Table 3.7: Transformer properties used in simulation.

160 kVA 10/0.4-kV transformer 250 kVA 10/0.4-kV transformer 400 kVA 10/0.4-kV transformer

3.5

Resistance Ω

Reactance Ω

Rated capacity kVA

0.0204 0.0126 0.0074

0.0675 0.0445 0.0291

160 250 400


This section investigates the relative accuracy of certain modeling and methodological assumptions on the results. First, simplifications to the distribution island model are analyzed, which aim to reduce computation time. A second part investigates the required temporal resolution of load data and grid simulation, in order to obtain accurate grid impact evaluation. Last, the stability of results with respect to the number of replications and design points is analyzed.

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3.5.1

Distribution island modeling approach

As mentioned in Section 3.4.2, simplification of the grid model could provide important reduction in computation time. One approach would be to assume feeders as independent units, eliminating the impact of other feeders in the same distribution island. This assumption might not hold when significant loads are present in neighboring feeders, creating voltage drop at the transformer. To study this effect, simulations of single feeders were compared with simulations of distribution islands with more feeders connected to the same transformer. Additionally, the dummy island approach was investigated, which represents an intermediate solution. With this approach, the feeder of interest is modeled in detail, while the remainder of the distribution island is represented by an aggregate load applied directly at the transformer, as illustrated schematically in Figure 3.9, and previously in Figure 3.7, on page 52. This approach has been used in similar studies before [3, 128, 202]. In distribution load flow analysis, more advanced lumping techniques may be used to represent lateral branches [51], which are however not explored here. More specifically, the same 1296 design points used for the main analysis were simulated for the three modeling approaches. Only one replication was used here, because full island simulations are very computationally expensive, and the study only aimed to compare the relative error. Therefore, the exact same loads were used for the three approaches. For single feeders, the same main feeder was simulated each time, ignoring loads in the remainder island. In the dummy island approach, based on the neighborhood type and values of NI , r HP,I and r PV,I , buildings were sampled from the stock, with their optional heat pump and PV. The respective loads were aggregated and applied at the transformer. For full island simulations, the same sampled buildings were used, but feeders were additionally created, where buildings were randomly positioned. Based on the value of NI , one to six additional feeders

Single

Dummy

Island

Figure 3.9: Modeling approaches investigated for the distribution island simulation.


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0.01

ΔUmax (pu)

-0.02

-0.04

0 -0.01 -0.02

-0.06

-0.03

0.1

0.1

0.08

0.08 ΔUmin (pu)

0.06

95%w

ΔUmin (pu)

ΔImax (pu)

0

0.04 0.02

Dummy 95 % of cases Dummy median Single 95 % of cases Single median

0.06 0.04 0.02 0

0

-0.02

-0.02 160

250 Sn (kVA)

400

160

250

400

Sn (kVA)

Figure 3.10: Differences in per unit current and voltage indicators for single and dummy island approaches, compared to full island simulations, per transformer rated capacity S n . Lines denote the median and shaded areas contain 95 % of data points, from percentile p 2.5 to p 97.5 . were generated, each containing between three and 45 buildings. The number of buildings per feeder was randomly sampled, with some rules to ensure the total NI is reached. As concerns cable lengths, the average distance between buildings l avg of 22.5 m for rural and 7.5 m for urban neighborhoods was used for all additional feeders. Furthermore, the strong cable option was used for those feeders. Figure 3.10 shows the per unit (pu) differences for the main indicators at feeder level, for single feeder and dummy island simulations, compared to the full island simulations, for different transformer rated capacities S n . Peak power or annual energy indicators are insignificantly impacted at the feeder level, with deviations below 1 % in all cases. For all current and voltage indicators, the error is larger for small transformers, because neighboring feeders create larger voltage drop, influencing more the results of the main feeder. The maximum current I max tends to be underestimated with the single feeder approach, while the dummy approach is on average accurate, with deviations within 0.01 pu for at least 95 % of cases with the small transformer. In the simulated cases,

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Dummy 90% of cases Dummy median Single 90% of cases Single median

0.1

ΔUmin (pu)

0.08 0.06 0.04 0.02 0 0

30

60

90

NI

120

150

180

0

20

40

60

80

100

rHP,I

Figure 3.11: Differences in Umin for single feeder and dummy island approaches compared to island simulation, per number of buildings NI (left) and heat pump penetration level r HP,I (right) in the remainder island. cable overloading was mainly due to high demand from heat pumps. The trend for I max can, therefore, also be reflected on the minimum voltage indicators, Umin and 95%w Umin . For the small transformer, single feeder simulations overestimate Umin by 95%w up to 0.1 pu, and the weekly minimum Umin up to 0.07 pu. In contrast, the dummy approach yields results very close to the full island simulations, with deviations below 0.03 pu and 0.02 pu respectively. Regarding Umax , deviations are smaller in absolute values, because voltage rise was generally more limited than voltage drop due to heat pumps. Single feeder simulations underestimated the extreme values of this indicator as well, while the dummy approach was on average correct, with very limited deviations. Figure 3.11 furthermore displays the deviations for Umin with respect to the number of buildings NI and percentage of heat pumps r HP,I in the remainder island. It is clear that an increase in both leads to larger deviation in Umin for the main feeder, especially when the island is not considered. In smaller islands, the deviation between modeling approaches will be reduced, as a result of less stress on the transformer. Also, for cases with very few heat pumps in the island (r HP,I ≈ 0%), the assumption of feeder independence would not produce significant inaccuracies. In all cases, the dummy island approach gives a rather good approximation of the full island results. Weighing the accuracy and savings in computation time offered by the dummy island approach compared to island simulation, the latter shown in Figure 3.12, it is concluded that this solution offers important benefits to probabilistic impact studies. This is also true because parametrization of a dummy load is simpler than that of multiple additional feeders, thus also limiting the number of options to


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1000 100 % of cases 90 % of cases 50 % of cases median

CPU time (min)

100

10

1

0.1 Island

Dummy

Single

Figure 3.12: CPU-time for annual simulations with the different modeling approaches, performed on a workstation with two Intel Xeon E5-2630 v4 2.2 GHz processors and 64 GB RAM, running on Windows 10 Pro 64-bit. simulate. While the single feeder approach results in even less cases, the accuracy was considered insufficient. For all these reasons, the dummy approach is chosen for all other analyses in this thesis.

3.5.2

Input data and simulation temporal resolution

In grid impact studies, network simulation resolution depends on the quantities of interest and the input data resolution, while trying to balance accuracy and computation time constraints. Regarding the studied indicators, we have seen in Section 3.2 that voltage levels need to be assessed at 10-min averages, while for thermal constraints, for instance cable and transformer loading, longer periods may be sufficient. Temporal resolution of boundary conditions defines, to a great extent, the accuracy of network simulations. As high resolution data are hard to find, for instance, smart meter data for long periods are often stored at intervals of 15 min or longer [61, 145], sensitivity analysis of grid impact indicators to input resolution is of great importance. Baetens et al. [14] have previously examined the influence of boundary condition resolution on power and voltage profiles of a specific net zero energy residential neighborhood. Due to simultaneity of loads in the grid, heat pump demand and irradiance data resolution were found to significantly impact the resulting profiles. In this work, grid impact is evaluated for many feeder cases based on few indicators, rather than entire time series. This section, therefore, examines the influence of input and simulation temporal resolution (∆t ) on those indicators.

60

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For this analysis, the same 1296 design points used for the main analysis are simulated for different temporal resolutions, with one replication per design point. Resolutions of 1, 5, 10, 15, 30 and 60 min are evaluated, 1 min being the finest resolution for which solar radiation and occupancy modeled data could be obtained. ∆t refers to both the grid simulation output resolution and the load profile resolution. Simulating the grid with smaller output step than the loads’ resolution would provide no additional information, since the program linearly interpolates the inputs. Load profiles are simulated at 1-min intervals, and averaged for the required ∆t . Due to memory restrictions for the building simulations, inputs concerning internal heat gains were read as 5-min averages. Given the slower dynamics of a building’s thermal response, this shouldn’t have a large impact on the heat pump electricity use. Last, different averaging periods dt for the maximum current I max and peak transformer load S max are also discussed. As explained in Section 3.2, while standards evaluate the 10-min averaged voltage [53], for I max and S max averaging periods between 10 and 60 min are examined in literature [20, 155]. In Figure 3.13 the differences in 10-min averaged extreme voltage are shown for all ∆t , with respect to values obtained from ∆t =1 min. For ∆t < 10 min, the indicators are extracted from 10-min moving averages, while for ∆t ≥ 10 min, the maximum or minimum of that profile is taken, since simple interpolation, constant or linear, would yield the same result. According to Figure 3.13, for resolutions below 15 min there are both positive and negative deviations, while for 30 and 60 min the minimum voltage is always overestimated, denoting an actual underestimation of the problem severity. Deviations for ∆t until 10 min remain below 0.04 pu, but become significant, up to 0.18 pu for p 97.5 of cases in hourly resolution. Taking into account the median and spread of the deviations for the minimum voltage, resolution lower than 10 or 15 min should be avoided. In contrast with Umin , deviations for the maximum voltage are much smaller, as voltage rise was generally limited for all cases. For ∆t below 10 min deviations are approximately centered around zero, after which a general underestimation is observed, when the resolution is too low to capture 10-min peaks in voltage. Contrary to the limited median voltage deviations discussed above, Figure 3.14 dt dt reveals large deviations for the maximum current I max and transformer load S max , depending on the combination of averaging period dt and resolution ∆t . Note that, a jump exists for ∆t =10 min and dt =15 min, because the indicators were first interpolated to 5-min intervals before being averaged to 15-min. As expected, the deviations increase with ∆t , and decrease for longer averaging period dt . For all averaging periods, the median deviations remain below 0.03 pu for both indicators when the resolution is 10 min or higher. However, over the entire range of simulations, larger deviations were observed, especially for ∆t = 60 min, not shown in this graph. The higher deviations were found for dt = 10 min, where p 2.5 reached for both indicators approximately -0.13 pu and -0.38 pu for ∆t =10 min and ∆t =60 min


0.18

0.18 0.15

95 % of cases median

0.15 0.12

ΔUmax (pu)

0.12

ΔUmin (pu)

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0.09 0.06

0.09 0.06

0.03

0.03

0

0 -0.03

-0.03 1 5 10 15

30

60

1 5 10 15

Resolution Δt (min)

30

60


0

0

-0.03

-0.03

ΔSmax (pu)

-0.06 -0.09 -0.12

dt

dt

ΔImax (pu)

Figure 3.13: Differences in 10-min averaged minimum and maximum voltage per resolution ∆t , with respect to 1-min resolution. Lines denote the median and shaded areas contain 95 % of data points, from percentile p 2.5 to p 97.5 .

dt=10 dt=15

-0.06 -0.09 -0.12

dt=30 dt=60

-0.15

-0.15 1 5 10 15

30


60

1 5 10 15

30

60


Figure 3.14: Median differences in maximum feeder current and transformer peak power with respect to 1-min resolution, per resolution ∆t . The different lines indicate the median deviation, for different averaging periods dt (in min). respectively. For the selected averaging period in this thesis of dt = 30 min (see Section 3.2), deviations were fairly low for ∆t ≤10 min, limited to 0.01 pu for the median of I max , and below 0.10 pu for 95 % of cases. Resolution higher than 15 min would be preferred for voltage evaluation anyway, as previously mentioned. Based on the results for voltage and loading, a resolution of at least 10 min and corresponding input data are found to be necessary for grid impact analyses, in terms of accuracy. Navarro et al. [155] also found that underestimation of voltage problems would occur for 30 and 60 min resolution, when studying the impact of PV

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CPU time (min)

100 100 % of cases 90 % of cases 50 % of cases median

10

1

0.1

0.1 1 5 10 15

30

60


Figure 3.15: CPU-time for annual simulations and different temporal resolution ∆t . Simulations were performed on a workstation with two Intel Xeon E5-2630 v4 2.2 GHz processors and 64 GB RAM, running on Windows 10 Pro 64-bit. on LV feeders in the UK. Assuming input data can be obtained for any resolution, in probabilistic assessment with numerous simulations, time constraints may inform the choice of a simulation resolution. The CPU-time required for annual simulation of the grids for different resolutions is given in Figure 3.15. As expected, lower resolution simulation requires also less computation time. The median CPU time is about 60 times larger for 1-min simulation compared to hourly simulation. In case of simulated loads, additional time should be foreseen for the profile generation as well. Furthermore, a disadvantage of high resolution simulations is the accordingly larger output file size, which can be an issue in probabilistic approaches with thousands of simulations. As previously mentioned, memory limitations may also arise while reading input data into the simulation program. Considering accuracy of results and available computation budget for this work, 5-min resolution was chosen for all other analyses.

3.5.3

Number of replications

As discussed in Section 3.3, several replications are produced for each design point, in order to take into account uncertain inputs to the simulation, such as the building properties and the location of loads on the grid. For each replication of every design point, different buildings with heat pumps and optional PV are sampled and assigned to random locations on the feeder. The aim is to replicate each design point enough times to approximate the distribution of results at those points as accurately as possible. Given that simulation time is not insignificant, it is desirable to reduce the number of replications to the least necessary. For certain inputs, like the annual


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electricity consumption, a good approximation of the distribution mean might suffice, while for other indicators, such as Umin , also the distribution tails are of interest. This section, therefore, investigates the accuracy obtained for different indicators and distribution statistics, depending on the number of replications m. The first part of the analysis focuses on load indicators at the feeder level, which do not require grid simulation, but only load aggregation, therefore allowing for a large m. One sixth of the 1296 design points are selected at random to create a smaller dataset for analysis. For each design point, taking into account construction quality, number of buildings and heat pumps, dwellings are sampled 1000 times for the feeder only. This dataset then comprised 216 design points with 1000 replications each. In order to evaluate the convergence of distribution statistics at each design point for different values of m, non-parametric bootstrapping is used. Bootstrapping consists in taking multiple samples of equal size drawn with replacement from a single original sample, with the purpose to estimate the sampling distribution of a statistic, such as the sample mean. The standard deviation of the bootstrap distribution std∗ approximates the sampling standard deviation of the statistic. As such, it gives an estimate of the standard error of the statistic, that is, how variable it is when different random samples of same size are taken from the population. Consequently, std∗ can be used to assess convergence in Monte Carlo simulations [105]. The bootstrap standard deviation of the sample mean, percentile or other statistic a can be calculated as follows [48]: sP B (a ∗ − a ∗ )2 i =1 i std∗ (a) = (3.1) B −1 where B is the number of bootstrap samples, here B =100, a i∗ is the computed P statistic from sample i , and a ∗ = B1 Bi=1 a i∗ is the averaged statistic over all bootstrap samples. One value for std∗ (a) is computed for each design point separately, and for each different number of replications m. For m smaller than the total available replications, 1000 here, a sub-sample of m replications is created with random sampling, without replacement, from the large set. To reduce the influence of this sampling, calculation of std∗ is repeated for a series of 10 different random subsamples. Three series of 10 runs are used in the first part of the analysis, for the load indicators. Last, the bootstrapped standard deviations std∗ are normalized by dividing with the mean from the large sample of 1000 replications, per design point, to produce the reported relative deviations. Here this value is used since the large sample was available. In cases where consecutive replications are added, the stopping rule could be based on the normalized deviation obtained by division with the mean of the current sample [105].

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Relative deviation (-)

10 0 Series 1 Series 2 Series 3 mean of series

100 % of cases 90 % of cases 50 % of cases median

10-1

10-2

Ed, mean 5

10 15 20

Ed, mean 50

100 200

Number of replications m

500 1000

5

10 15 20

50

100 200

500 1000


Figure 3.16: Bootstrap relative standard deviations for the sample mean of the annual feeder electricity consumption E d , per number of replications m . Left: for a specific design point, results for three series of 10 runs, lines denoting the mean per series. Right: distribution over all design points of the mean deviation from three series. Bootstrap samples B = 100. The left panel of Figure 3.16 provides an example of deviations for the annual electricity consumption by feeder households E d and one specific design point. Here, the bootstrapped relative deviation is shown for all 10 runs of three series, with lines denoting the mean for each m. The three series give equivalent results, therefore, the mean deviation among all three is further used as indicative for each design point. This allows presenting the result for all design points graphically, such as on the right panel of Figure 3.16. This plot shows the distribution of mean relative deviation over the dataset of 216 design points. As expected, the deviation drops with increasing m, reaching an average minimum of 1.7 % for 1000 replications. This means that, for a given design point, repeated random sets of 1000 replications would give a mean for E d with a precision (standard error) of 1.7 %. The median standard error is around 20 % for just 5 replications, and drops to 11 % for 20 replications. If higher accuracy is required for approximation of the electricity consumption of each individual feeder case, even more replications would be necessary. Figure 3.17 shows the relative deviations for the feeder peak demand, based on the same data and procedure. In the left panel, the sample mean is analyzed, showing smaller relative deviations than for E d , below 10 % already from 10 replications. Given that for peak loads extreme values are generally of interest, the 90th percentile is furthermore analyzed in the right panel of Figure 3.17. We may notice that deviations are generally higher, with larger differences between design points, seen by the wider spread of the distribution. As a general rule, more replications are necessary to obtain more precise distribution tail statistics [220].


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Relative deviation (-)

10 0 100 % of cases 90 % of cases 50 % of cases median

10-1

10-2

10-3

Pd, mean 5

10 1520

Pd, p90 50

100 200

500 1000

5

10 1520


50

100 200

500 1000


Figure 3.17: Bootstrap relative standard deviations for the feeder peak electricity demand P d , for all design points, per number of replications m . The mean deviation from three series of 10 runs is given per design point. Left: sample mean. Right: 90th percentile p 90 . Bootstrap samples B = 100.

4 3

Z-scores of Pd

2 1 0 -1 -2 -3 -4 1

2

5

10

15 20

50

100

200

500

1000


Figure 3.18: Distribution of the normalized feeder peak demand P d over all design points, per number of replications m . Each box-plot contains all m replications per design point, which are randomly chosen from the available 1000. The central rectangle spans from percentile p 25 to p 75 , whiskers extend to p 5 and p 95 , the middle bar marks the median, and × the mean.

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It should be emphasized that the above results investigate the precision in distribution approximation within each design point. This is necessary when conclusions for a very specific case need to be drawn. However, when looking at the entire dataset, less replications are needed. For instance, Figure 3.18 shows the distribution of P d for all 216 design points, depending on the number of replications m. We may see that the overall distribution remains rather stable, even for as few as five replications per design point. In the second part of this analysis, the influence of m on the simulated grid indicators is examined using the previous approach, but with more design points and less replications. Here, the entire dataset of 1296 design points is replicated 50 times, and only one series of 10 bootstrapping runs is used to analyze convergence of the distribution statistics. Furthermore, instead of relative deviations, here indicators are normalized by dividing with the nominal voltage or rated capacity of the loaded components as appropriate, presenting standard errors in per-unit (pu) values. Figure 3.19 summarizes the bootstrap standard errors for the minimum voltage Umin , maximum current I max and maximum transformer load S max . Except for the mean, standard errors are also calculated for the 10th or 90th percentile, to assess precision of the distribution tail. We may note steady reduction of the standard errors with increasing m for the mean indicators, while fluctuations and slower convergence are observed for the tail percentiles. For all indicators, we furthermore notice large variation among design points, because larger feeders and islands have more variability in their loads and network response. In terms of error values, for the majority of cases (p 95 : upper limit of area that includes 90 % of data), 5 replications give already standard errors almost below 0.02 and 0.03 pu for the mean and p 10 of Umin respectively. For I max and S max , the median errors are much larger, but the maximum errors are about double that of Umin , because the former have larger spread. However, the significance of 0.01 pu error is different in each context. Since voltage should be within 0.9 and 1.1 pu for normal conditions, we might consider 0.01 pu, or 5 % of this range, to be a significant deviation. On the contrary, overloading of distribution transformers, for example, is acceptable even up to 3 pu for short periods of time [97], where 0.01 pu seems unimportant. Generally, in order to select the minimum required number of replications, the level of precision should be specified in advance, also determining the percentage of cases it should cover, since we are looking globally on a whole range of cases. This depends, of course, on the quantities of interest and purpose of the analysis. For instance, obtaining p 10 of the Umin distribution with a precision of 0.02 pu for 95 % of the cases would require 50 replications, as per Figure 3.19. The same precision for the mean would only need 10 replications. Furthermore, for different indicators other minimal requirements may be appropriate. Since the total required simulation


10

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67

0 100 % of cases 90 % of cases

-1 10

50 % of cases median

-2 10

-3 10

Standard error (pu)

-4 10 0 10

Umin, mean

Umin, p10

Imax, mean

Imax, p90

Smax, mean

Smax, p90

-1 10

-2 10

-3 10

-4 10 0 10

-1 10

-2 10

-3 10

-4 10

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50


Figure 3.19: Bootstrap standard error for the sample mean and percentile of grid indicators, per number of replications m . The distribution over all 1296 design points is shown, calculated as the mean from a series of 10 runs with B = 100 bootstrap samples.

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0.95

Umin (pu)

0.9 0.85 0.8 0.75 0.7 2

5

10

15

20

25

30

35

40

45

50


Figure 3.20: Distribution of the minimum voltage Umin over an example selection of cases, per number of replications m . The selection includes rural moderate feeders with less than 30 buildings, connected to a 250 kVA transformer with Uref at 1 pu (total of 24 design points). Each box-plot contains all m replications per design point, which are randomly chosen from the available 50. For explanation of the box-plot, see Figure 3.18. time increases linearly with the number of replications, computation budget should be also considered when selecting m. More sophisticated approaches in literature may optimize the use of simulation budget, by allocating a different number of replications per design point, depending on the local variance [2, 123]. In this thesis, 20 replications were used for the main investigation, taking into account computation time. According to the present analysis, this m results in standard errors for 95 % of cases and the 10th or 90th percentile below 15 % for electricity consumption and peak demand, below 0.06 pu for I max , and 0.03 pu for Umin and S max . It should be noted once more, that these are the standard errors expected for the distribution within each design point. The distribution over the entire design space, or some subset of it, is better approximated with fewer replications. As an example, Figure 3.20 shows the distribution of Umin for a small subset of 24 design points is well approximated with 10 replications. This section analyzed how the number of replications influences the distribution statistics of certain grid impact indicators. A somewhat different question regards the performance of metamodeling for different numbers of available replications. This question will be addressed in Chapter 6.


3.5.4

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Number of design points

The previous section studied how the number of replications m influences the results, for a fixed large number of design points. Here, the inverse problem is studied, varying the number of design points n, for 20 replications per design point. Both analyses aim to investigate whether the number of simulation runs n × m can be reduced, to limit computation time. Subsets of the large dataset with 1296 design points are taken, with n ∈{27, 54, 108, 216, 432, 648, 864, 1080, 1296}. Since the Sobol’ sequence adds points incrementally, the subsets include the first n points each time. In this case, the standard bootstrap method may not give accurate error estimates, since the design points generated with the Sobol’ sequence are not independent. For more reliable error estimation, data points could be added in sets of independent Sobol’ sequences, generated with randomization techniques. Then the error can be estimated as the standard error of the sample statistic of interest calculated from the different sets [89, 90]. Here, however, since the large dataset was already available, the bootstrap approach is nevertheless used, to provide an indication of the relative impact of sample size for different indicators and sample statistics. In this section, the convergence of the entire dataset distribution is studied for different indicators and number of design points. The bootstrap standard deviation std∗ is calculated with Equation 3.1, and it is evaluated for the distribution mean and a percentile, depending on the indicator. In this analysis, only one value for std∗ is obtained per indicator and number of design points for the whole dataset, unlike in the previous section, where one value was taken per design point. That was the case because the distribution within each design point was analyzed. Here, the evaluated quantities are the mean or percentile per design point, calculated from 20 replications. For the minimum voltage, the mean and 10th percentile are used as indicators, while for other quantities the mean and 90th percentile. For each indicator, the bootstrapped standard deviations std∗ are calculated for both the mean and the same percentile of the entire dataset’s distribution. A series of 10 runs with B = 100 bootstrap samples is performed, and the average std∗ among runs is reported. The choice of 100 bootstrap samples is based on Kleijnen [113, p.97]. For voltage indicators, std∗ is expressed per unit (pu), while for others it is normalized by dividing with the sample mean from the large set of 1296 design points. The normalized errors are very similar also when normalizing with the sample mean of the current set of size n. Figure 3.21 summarizes the bootstrap standard errors for six quantities (see Table 3.1 on page 30). For each quantity (different plots), the mean and percentile per design point are evaluated as indicators (solid vs. dashed lines), both for the mean and a percentile of the global dataset distribution (black vs. orange lines), to assess precision of the distribution tail at both levels. We may note steady reduction of the standard errors with increasing n for all indicators. An exception is the error for

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Standard error (-)

100

-1

10

Mean of design point p90 of design point


1% or 5%

Et,d,net

Et,bf

100

Standard error (-)

Mean of dataset p90 of dataset

10-2

10-3

10-1





10-2

10-3

Imax

Smax

100

Standard error (pu)


-1

10





0.01 pu or 0.02 pu

0.01 pu or 0.02 pu

10-2

10-3

Umin 27 108 216

Umax 432

648

864

1080

1296 27 108 216

432

648

864

1080

1296

Number of design points n

Figure 3.21: Bootstrap standard error of grid indicators (Table 3.1 on page 30), per number of design points n , for the dataset mean and a percentile (black vs. orange). As indicators, both the mean and a percentile per design point are used (solid vs. dashed lines), calculated based on 20 replications. Each line represents the mean standard error from a series of 10 runs with B = 100 bootstrap samples each.


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p 90 of E t,bf at 27 design points, which is lower than that for 54 points (upper right subplot). This is because the set of 27 points under-represented the dataset, not including large values, thus resulting in smaller standard error as percentage of the large set mean. Generally, the errors for the mean or percentile indicators are very similar (continuous vs. dashed lines), except for the voltage, where the percentile indicators have larger errors than the mean indicator. This may be due to the more skewed distributions for voltage quantities within design points, compared to annual energy quantities. In all cases, the dataset mean is captured more accurately than the percentiles (black vs. orange lines), with quite large differences in standard errors between the two. Furthermore, voltage indicators have generally lower standard errors (in pu but also as percentages), while the backfeeding annual electricity E t,bf and the maximum current I max have the largest. The actual value of the standard errors should be considered with caution, given that the data points were not generated with random sampling. Nevertheless, we may compare these calculated errors with potential thresholds of required accuracy, represented by the horizontal dashed lines in Figure 3.21, namely 1 and 5 %, and 0.01 and 0.02 pu. For specific applications, such thresholds may be set per indicator, which could be then used to determine the required number of design points, based on a procedure as described in [90]. Here, the calculated bootstrap errors for Umax are below both thresholds even for the smallest dataset. For Umin the more demanding indicator, that is p 10 per design point and the dataset, requires more than 648 design points for an error below 0.02 pu. For the other indicators, 1 % accuracy is not reached, even for the distribution mean with the large dataset. For more lenient thresholds at 5 %, a dataset of 432 design points would suffice for the dataset mean (black lines) of all indicators. However, for the dataset distribution tail (orange lines), at least 864 points are needed for most indicators, while for E t,bf 5 % is not even achieved for 1296 points. While the above standard errors refer to the entire dataset distribution, if we’re interested to obtain a precise distribution for subsets of the dataset, more design points would be needed. As an example, Figure 3.22 shows the distributions of the mean annual electricity net consumption E t,d,net and 10th percentile minimum voltage Umin for a small subset. This subset, also used in Figure 3.20, represents rural moderate feeders with less than 30 buildings, connected to a 250 kVA transformer with Uref at 1 pu, and is represented by 24 design points in the large dataset. We may observe that for a total design size n below 108 this case is not represented at all in the dataset. As n increases, more design points are added to this subset. If we’re interested in these specific cases, a dataset of size 432 is necessary to even obtain more than two cases. For a better approximation of the distribution, n = 864 would be needed for both depicted indicators. This short analysis gives an indication of how the sample size might impact the resulting indicator distributions. It was shown that this impact may vary per quantity

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1

Umin, p10 (pu)

0.95 0.9 0.85 0.8

Et,d,net, mean (z-scores)

0.75 3 2 1 0 -1 -2 -3

00

1

27 54 108

2

6

10

15

22

24

216

432

648

864

1080

1296

Number of design points n (total/in subset)

Figure 3.22: Distribution of the normalized mean E t,d,net and the 10th percentile of the minimum voltage Umin over an example selection of cases, per number of design points in the whole dataset. The selection includes rural moderate feeders with less than 30 buildings, connected to a 250 kVA transformer with Uref at 1 pu. The available design points to create each box-plot are given on the x-axis as well. The central rectangle spans from percentile p 25 to p 75 , whiskers extend to p 5 and p 95 , the middle bar marks the median, and × the mean. of interest and distribution statistic. In order to reduce the number of design points n to be simulated, sequential addition of design points could be investigated, where standard errors are evaluated when new sets of points are added [90]. For this purpose, the specific objectives and precision thresholds should be specified for different indicators. On the other hand, instead of analyzing the distribution of an indicator itself, one may be interested in the performance of a metamodel built using this dataset. This will be investigated in Section 6.4.4. For the remainder of this thesis, since multiple different analyses are made, the large dataset of 1296 design points is used.

Strengths and limitations of the methodology

3.6

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73

Strengths and limitations of the methodology

In Section 3.1, the general approach and assumptions of this framework have been discussed. Models and methods were chosen that represent buildings and the electrical network in detail, with the purpose to provide accurate grid impact evaluation and allow to investigate the influence of building characteristics. The major strength of this method is that it allows a probabilistic assessment, that takes into account uncertainties in the loads, for many different network sizes and characteristics. In this way, better informed decisions can be made when it comes to policy assessment. Furthermore, the method is flexible, in the sense that different LCTs and boundary conditions can be incorporated. Limitations of this approach are mainly due to computational restrictions. Computation time is large both because of long individual simulations, and because many simulations are needed to cover the design space. As regards reducing the simulation time, some simplifications have been implemented, such as decoupling building and grid simulations, and modeling other feeders of the distribution island as an aggregate load (dummy approach). The effect of the latter has been investigated and shown to be small. Decoupling of the building and grid simulations could be acceptable, so long as no communication exists between the two. This may not be true in future grids, where technology could allow the implementation of network-driven demand side management (DSM). As such, the present approach is restricted to representing limited interaction scenarios. In future work, code improvements, and high performance computing could be explored to reduce computation time [109, 173]. With respect to the number of required simulations, the more options we wish to investigate, the larger the problem’s dimensionality becomes. With increasing dimensionality, the amount of data needed for reliable results grows exponentially. In combination with time consuming simulations, an inevitable restriction of the scope is required. As a result, the present work has been limited to investigate residential neighborhoods with single-family dwellings, and only two LCTs. In order to implement other technologies and cases, additional information or assumptions would be needed, that would define specific combinations to be investigated, instead of the entire range of possibilities. Furthermore, there is room for improvement of the experimental design, which could reduce the number of required simulations. Similarly, the number of replications could be lowered, depending on the required accuracy and purpose of the analysis, as illustrated in a previous section. Last, the model complexity and detail of this approach prohibit its use by nonexperts. This is the reason why in the second part of this thesis metamodeling is proposed, which should provide both faster and easy to use representations of the results obtained through the detailed probabilistic framework.

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3.7

Probabilistic framework: Recapitulation

This chapter introduced a probabilistic framework that allows investigating issues at the LV grid level, arising from the integration of LCTs in residential buildings. The same framework serves to generate a dataset for metamodeling grid impact indicators, which is necessary in order to utilize the results in high-level studies. To reduce the problem dimensionality, the scope was limited to Belgian residential neighborhoods with single-family dwellings, potentially equipped with air-source heat pumps and rooftop PV systems. Quantities of interest were then identified, which are useful in evaluating the impact of LCTs on load profiles and the resulting strain on the grid. Indicators were defined pertaining to voltage quality, grid component overloading, peak and total electricity exchange with the supply system. The probabilistic framework evaluates those grid impact indicators by propagating uncertainties from inputs to outputs using the Monte Carlo approach. Grid parameters, uncertain factors and scenarios are considered in an experimental design that selects the feeder cases to be examined and later used for metamodeling. Uncertain factors representing variability in building characteristics are taken into account by first creating a building stock, which comprises demand and generation profiles. Those are then sampled in the feeder cases. Both the loads and network response are simulated using detailed building and network models, implemented in Modelica language. The models and methods were described in this chapter. Given the increased computational demand of a probabilistic approach, the last part of this chapter investigated several methodological assumptions that aimed to reduce calculation time. Specifically, grid simplification by representing the distribution island with an aggregate load was shown to cause little loss of accuracy compared to full island simulations, allowing important reduction in simulation time. Furthermore, the influence of temporal resolution was investigated, indicating that for better assessment of voltage indicators, 10-min input data are required. Moreover, comparison of the obtained accuracy for different distribution statistics and indicators was performed, for different numbers of replications. For this work, 20 replications were found to produce accurate enough representation of the indicator distributions for each individual feeder case. Last, the impact of sample size on the resulting indicator distributions was analyzed. The required number of design points was shown to vary greatly between indicators and accuracy thresholds. For applications with specific accuracy requirements, sequential addition of sets of points while monitoring convergence would be valuable. This framework aims to both analyze the grid impact of LCTs, and to provide a basis for further metamodeling of important grid indicators. The next chapter presents a thorough analysis of the results obtained by implementation of the framework on Belgian residential LV grids, while metamodeling is discussed in Chapters 5 and 6.

4

Impact of heat pumps and PVs on Belgian residential feeders

This chapter presents the simulation results from application of the probabilistic framework to a range of cases designed as Belgian residential low-voltage (LV) grids, where heat pumps and PV are introduced. Details of the models and inputs were given in Chapter 3, where the scope was furthermore defined. The analysis in this chapter aims to identify the main technical bottlenecks to the integration of heat pumps and PV in the LV feeders, and to examine the impact of grid and building parameters on those results. A general overview of the results is first given, followed by a sensitivity analysis that identifies important grid parameters. Neighborhood load profiles are then analyzed, putting into perspective the importance of demand diversity in terms of building characteristics and load simultaneity. The following sections are dedicated to overloading problems and voltage violations, where a probabilistic assessment is presented. The role of building characteristics is furthermore investigated, providing insight into the potential of building retrofitting to reduce grid impacts. The last three sections deal with the sensitivity of the presented results. They focus on weather scenarios, technological scenarios related to heat pump power factors, and variations in occupant modeling.

4.1

Overview

This analysis starts with an exploration of annual simulation results in Figure 4.1. These plots are made for an example grid case, where both voltage problems and transformer overloading occurred. This feeder is one of the largest considered and with rather high penetration level of heat pumps and PV, therefore, it is a good example to illustrate the severity of potential issues. From the top left plot of Figure 4.1 we may observe the seasonal pattern of both PV generation and heat pump loads, the latter seen in the total load curve, since

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150

Total load Base-load PV

Feeder load (kW)

100 50 0 -50 -100

1.1 pu

1.1

Voltage (pu)

1 0.9

0.85 pu

0.8

Phase A Phase B Phase C

0.7 1 pu

1

Phase A Phase B Phase C

Current (pu)

0.8 0.6 0.4 0.2

Transformer real power (pu)

0 1 pu

1 0.5 0

-0.5 0

50

100

150

200

Time (days)

250

300

350

10

11

12

13

14

15

16

17

Time (days)

Figure 4.1: Time-series annual results for example case of rural new neighborhood, weak feeder with N =41, r HP =77%, r PV =53%, remainder island with NI =33, r HP,I =52%, r PV,I =56%, and 160 kVA transformer with Uref =1 pu (see Table 3.5 for parameters). From top to bottom: Feeder-level loads, voltage at the feeder furthest node per phase, current in the first feeder segment per line, and total transformer load (real power). Profiles averaged per 30-min, except for voltage where 10-min average is shown. Left: entire year. Right: zoom into the coldest week.

Overview

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77

the base-load remains largely unchanged throughout the year. The right hand side detail furthermore shows the relative contribution of this base-load to the total load is small in a cold winter week, given that 77 % of buildings are heated with a heat pump. Despite the 53 % penetration of PV, generation in the same week is far from sufficient to cover demand. However, during summer, generation exceeds demand, resulting also in reverse power flow at the transformer (bottom left). The same seasonal pattern can be observed on the transformer load (bottom), but also on voltage levels and current in the feeder. The details on the right show that discrepancies may exist between the different phases. For example here, phases A and B seem to have more heat pumps connected to them, resulting in lower voltage and higher current in winter. Furthermore, phase B appears to have more overvoltage problems in summer, possibly because of more PV systems connected to it. For this example grid case, the transformer is overloaded in the coldest week for periods of considerable duration, while the cable of phase A is briefly close to the limit. Furthermore, the voltage level in phases A and B drops far below the limit of 0.85 pu on many occasions. The upper limit of 1.1 pu is violated in few occasions during summer for phase B, but for this feeder undervoltage is the major problem. The previous detailed results focused on one specific network with important voltage and overloading issues. The entire simulated dataset consists in 1296 feeder cases, each simulated for 20 replications, where buildings were randomly sampled and assigned to the network. To provide a first evaluation of the overall results, Figure 4.2 shows the probability density functions for a selection of indicators. Kernel density estimation with normal kernels was used to estimate the distributions. Separate distributions are presented for different subsets of the original dataset. Subsets were selected to illustrate the effect of an important grid parameter for each indicator, thus providing a more informative graphical representation. A deeper analysis of the influential parameters will follow in the next sections. On the top of Figure 4.2, the annual net electricity consumption E t,d,net and backfeeding E t,bf are presented per building (p.b.), to eliminate the influence of distribution island size. There is a clear large impact of the heat pump penetration level on the net consumption per building. On average, the latter approximately doubles for r HP,t above 75 % compared to below 25 %. These values refer to the net demand, thus having taken into account self consumption from PV and energy losses in the grid. Furthermore, these results are from the extreme weather scenario used for most of the analysis. Section 4.7 will show how these values vary for different weather profiles in detail. The electricity consumed by households in a feeder (E d ) is shown on the second row, also per building in the feeder. The base-load annual demand for the simulated 300

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rHP,t ϵ[0, 25) rHP,t ϵ[25, 50) rHP,t ϵ[50, 75) rHP,t ϵ[75, 100]

0

2

4

6

8

rPV,t ϵ[0, 25) rPV,t ϵ[25, 50) rPV,t ϵ[50, 75) rPV,t ϵ[75, 100]

10

-3

-2.5

Et,d,net (MW h/year p.b.)

-2

-1.5

-1

-0.5

rHP ϵ[0, 25) rHP ϵ[25, 50) rHP ϵ[50, 75) rHP ϵ[75, 100]

0

4

2

6

0

Et,bf (MW h/year p.b.)

8

T = Rural T = Urban

10

0

1

2

Ed (MW h/year p.b.)

3

5

4

7

6

8

Pd (kW p.b.) rHP,t ϵ[0, 25) rHP,t ϵ[25, 50) rHP,t ϵ[50, 75) rHP,t ϵ[75, 100] 1 pu

-5

-4

-3

-2

0

-1

1

3

2

5

4

Pt (pu) rHP,t ϵ[0, 25) rHP,t ϵ[25, 50) rHP,t ϵ[50, 75) rHP,t ϵ[75, 100] 1 pu

0

1

3

2

5

4

rHP ϵ[0, 25) rHP ϵ[25, 50) rHP ϵ[50, 75) rHP ϵ[75, 100] 1 pu

6

0

0.5

1.5

1

Smax (pu) T = Rural T = Urban 0.85 pu

0.4

0.5

0.6

2.5

2

Imax (pu) Uref = 0.95 pu Uref = 1 pu Uref = 1.05 pu 1.1 pu

0.7

0.8

0.9

Umin (pu)

1

1.1

1.2

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Umax (pu)

Figure 4.2: Probability density functions of grid impact indicators, based on kernel density estimation (normal kernel). Different line styles represent subsets of the data, as indicated in subplot legends. Some indicators are presented per building (p.b.) to eliminate the effect of grid size, while others relative to nominal limits (pu). For explanation of symbols see Table 3.1 on page 30 for indicators, and Table 3.5 on page 53 for grid parameters. Here, P t is the transformer signed maximum absolute load (demand or back-feeding).

Overview

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79

households is on average 3.3 MWh. This value happens to coincide with the average reported consumption per consumer in Flanders for 2017 [187]. The latter, however, includes electricity uses that were not considered in the model, such as direct electric heating, as well as few non-domestic consumers. On the other hand, the baseload electricity consumption model is based on occupant behavior and appliance power requirements dating before 2010 [13]. Change of behavior, improved energy efficiency and new appliances have since lowered the average consumption, as also reported for Flanders [187]. Therefore, more research would be required to represent more accurately future electricity consumption patterns, with respect to household appliances and lighting. Compared to potential base-load variations, inclusion of heat pumps in the feeders alters the total consumption much more drastically. For this extreme weather scenario, cases with more than 8 MWh/year per building may be observed for very high penetration of heat pumps. While efficiency tends to reduce electricity consumption every year for appliances and lighting, other much larger uses should be added as well, such as electric vehicles. While E t,d,net in Figure 4.2 has shown the net consumption to be lower than the demand, it remains significant, as PV production is not sufficient to cover feeder needs on an annual basis, with the limited capacity of 5 kW per connection. The lack of simultaneity between demand and production furthermore leads to significant amounts of backfeeding electricity E t,bf , with backfeeding occurring already for low levels of PV penetration. In order to reduce the burden on the medium-voltage (MV) grid and overall electricity distribution system, future grids should aim to better balance loads and local generation at the neighborhood level. The peak demand P d , on the second row of Figure 4.2, represents the expected maximum demand per building, having taken the diversity between consumers into account, but not including any self consumption. Rural feeders have noticeably larger peaks, because detached dwellings generally have greater heating requirements compared to terraced. P d covers a wide range depending on the heat pump penetration level, but is in general significantly higher than currently assumed peak loads, averaged for many households. In this thesis, the simulated 300 households yield an after-diversity peak of 0.92 kW. Synthetic load profiles, used in the electricity market in Belgium, would yield, for an average household, a peak around 0.8 kW, assuming an annual electricity consumption of 3.5 MWh [213]. The synthetic profiles for consumers with higher night-time consumption—often caused by use of electric heating elements—would yield a peak around 1.3 kW for the same annual consumption, still on the low side of expected peaks when heat pumps are present. Evidently, uncoordinated plug-in electric vehicle charging in residential grids would add an even more significant stress on the grid. More details on the peak demand will be given in Section 4.3. With the mentioned demand peaks, and taking self consumption into account, P t depicts the signed peak load at the distribution transformer (absolute maximum

80

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load, positive or negative). This plot shows the shift from negative peaks (backfeeding) to positive peaks (demand) as heat pump penetration increases. For more than 75 % heat pumps, regardless of PV penetration the demand peaks are always higher than back-feeding peaks, indicating the stronger impact of heat pumps on grid loading. S max , expresses the maximum apparent power flowing through the transformer, which is generally very similar to the real power for mostly resistive loads. This indicator determines the maximum transformer loading, based on 30-min average values. While overloading is a thermal problem that depends on other factors, halfhour loading above 1 pu is considered in this thesis as a sign of potential overloading. As seen in this graph, many of the simulated cases exceed this limit. About 62 % of overloading occurrences relate to the 160 kVA-transformer, but even the larger 400 kVA-transformer is insufficient for 5 % of the cases where it is used . Thermal loading problems can be also observed for feeder cables, where the maximum current I max exceeds 1 pu. This problem is much less frequent than transformer overloading in the simulated cases. It only occurs in 4.7 % of all cases, mostly for heat pump penetration levels above 50 %. More details on overloading issues are given in Section 4.4. The bottom plots of Figure 4.2 show the distribution of minimum and maximum observed voltages. An important determinant for those distributions is the assumed reference voltage at the transformer, indicating that transformers with on-load tap changers for voltage control could provide important improvement. Generally, voltage drops are much more pronounced than overvoltage, suggesting heat pumps cause a larger problem than PV in the simulated cases. The minimum voltage Umin falls much below the prescribed limit of 0.85 pu for many cases, with more severe problems in rural feeders, where both the heat pump loads are higher, and the distances longer. These problems mostly occur in cold winter days, when heating demand is high, auxiliary heating is activated and PV production is minimal. On the other hand, the maximum limit at 1.1 pu is exceeded mostly for cases with increased reference voltage Uref . Overvoltage problems have stronger dependence on solar irradiance during summer, and on random coincidence of loads. This overview has shown there are severe impacts to be expected from high penetration levels of heat pumps and PV in LV residential grids, with larger impacts on rural feeders. The following sections will analyze in more detail different aspects influencing these results. To complement the observations made here, the next section performs a sensitivity analysis to formally determine the most influential grid parameters for each indicator.

Grid parameter sensitivity

4.2

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81


The previous section already gave an idea of some influential grid parameters for few indicators. Here a sensitivity analysis is performed for all grid parameters and a more comprehensive list of indicators. The aim is to understand how grid parameters impact each indicator, and to identify the important ones, both for result analysis in the next sections, and for metamodeling in Chapter 6. Several methods exist for sensitivity analysis, however literature has highlighted the benefits of global methods, as they better explore the input space and consider interactions. These methods include, among others [101, 196, 216]: d screening, such as the Morris method; d regression-based methods, for example standard regression coefficients and (partial) (rank) correlation coefficients; d variance-based methods, such as Sobol’ and Fourier Amplitude Sensitivity Test; and d metamodel-based methods, where a metamodel substitutes the original model, allowing the computation of variance-based sensitivity indexes at a lower evaluation cost. Variance-based methods are model independent and more informative, as they include higher order effects in their decomposition of the output variance. However, they are also computationally expensive and more complex to implement. In this work, regression-based methods are used, since they can be performed on the available dataset without need for additional model evaluations. Correlation coefficients are computed based on the available simulation results, to evaluate the importance of different grid parameters. The Pearson correlation coefficient is a sensitivity measure for the linear correlation between an input parameter and the output, and can be computed as: n P

ρx y = s

(x i − x)(y i − y)

i =1 n P

n P (x i − x)2 (y i − y)2 i =1 i =1

(4.1)

where x and y are two random variables, such as the input parameter and output, and n is the number of samples in the dataset. ρx y takes values between -1 and

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1, with 0 indicating no linear correlation, -1 and 1 indicating a perfect negative or positive linear correlation respectively. Instead of using this coefficient when the relationship is not linear but monotonic, the correlation coefficient can be calculated with Equation 4.1 based on the rank transformed data instead (i.e. replacing the values with their ranks when they are sorted), what is called Spearman’s or rank correlation [79]. Then the coefficient determines the strength of a monotonic correlation, taking also values between -1 and 1. The calculated coefficients with the above methods may be misleading in the presence of confounding variables z. These are, for instance, other grid parameters which are correlated with x. An example, in our case, is the neighborhood type T , and the average cable length l avg . In this case, it is preferable to use a partial correlation coefficient ρx y|z , which takes into account the effect of control variables z. ρx y|z is computed with Equation 4.1, but instead of x and y, the residuals from the following linear regression models are used: X X xˆ = α0 + α j z j and yˆ = β0 + β j z j (4.2) j

j

In the case where z is a single variable, the calculation is simplified to: ρx y − ρxz ρy z ρx y|z = q 2 )(1 − ρ 2 ) (1 − ρxz yz

(4.3)

with ρx y , ρxz and ρy z the simple correlations computed from Equation 4.1. The rank transformation can be also applied to partial correlation coefficients. For the results presented in Figure 4.3, the Pearson partial correlation coefficients were calculated per indicator, where for each grid parameter the others were taken as control variables. The linear correlations were similar to the respective rank correlations, albeit somewhat smaller for all cases. This would suggest potential nonlinear relationships. As more conservative, the Pearson coefficients are shown here. In Chapter 6, metamodels are built which can take into account also higher order effects. For indicators calculated solely on aggregation of electricity profiles, without grid simulation, the irrelevant parameters are excluded. Figure 4.3 is generally in agreement with the expected relationships. Demand indicators, such as E d and P d are not influenced by PV penetration levels r PV , while generation indicators, such as E g and P g are only influenced by the number of buildings and r PV . Indeed, the construction quality was not involved in sampling orientation and size for the PV systems. As regards the neighborhood type, the roof size restriction only reduced the average installed capacity of terraced houses—


Indicators

Pearson's partial correlation coefficient E t,bf

0.00

-0.09

-0.21

0.01

-0.47

-0.64

0.10

-0.80

-0.00

-0.00

-0.01

-0.00

E t,d,net

-0.20

-0.16

0.64

0.47

-0.17

0.96

0.79

-0.49

-0.01

-0.01

-0.01

0.03

Smax

-0.25

-0.11

0.49

0.48

0.15

0.90

0.74

0.17

-0.02

-0.01

0.02

0.02

Pt

-0.13

-0.11

0.15

0.26

-0.19

0.33

0.56

-0.44

-0.00

0.02

-0.02

0.03

P t,d,net

-0.29

-0.15

0.53

0.55

0.04

0.91

0.81

-0.08

-0.01

0.00

0.01

0.03

P t,bf

-0.00

-0.11

-0.29

-0.02

-0.48

-0.73

0.03

-0.81

-0.01

-0.01

-0.02

-0.00

I max

-0.25

-0.41

0.81

0.74

0.07

0.05

0.03

-0.04

-0.03

-0.22

0.02

-0.00

U min

0.07

0.13

-0.70

-0.44

0.01

-0.18

-0.10

0.06

0.16

0.64

-0.34

0.22

U 95%w min

0.10

0.18

-0.68

-0.47

0.01

-0.24

-0.15

0.05

0.20

0.79

-0.32

0.20

U max

-0.00

0.04

0.78

0.09

0.48

0.16

-0.00

0.21

-0.15

0.94

0.43

-0.29

Ed

-0.42

-0.46

0.95

0.83

0.02

E d,net

-0.42

-0.46

0.93

0.82

-0.52

Eg

-0.01

-0.05

-0.86

-0.03

-0.93

E bf

-0.03

-0.08

-0.71

0.07

-0.91

Pd

-0.58

-0.51

0.86

0.86

0.01

Pg

0.01

0.06

0.86

0.04

0.93

T

Q

N

r HP

r PV

|

83

1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00

NI r HP,I r PV,I Grid parameters

Sn

U ref

lavg

Ca

Figure 4.3: Sensitivity of grid impact indicators to the grid parameters, expressed as Pearson’s partial correlation coefficients. For each grid parameter, the others are taken as control variables. Insignificant results, with p -values above 0.05, are shown in gray font. For explanation of symbols see Table 3.1 on page 30 for indicators, and Table 3.5 on page 53 for grid parameters. predominant in urban feeders—to 3.85 kW, compared to 4 kW for detached houses— predominant in rural feeders. As a result, the neighborhood type did not have an important impact. Different allocation of PV capacities, for example based on building size or electricity demand, would result in higher generation for rural feeders, because detached dwellings are generally larger. Net consumption (E d,net and E t,d,net ), as well as the signed transformer peak load P t are strongly correlated with heat pump penetration levels r HP , but also with r PV , to a smaller extent. The correlation with r PV exists primarily because of cases with few heat pumps, where the peak load is negative (backfeeding), rather than because of limiting the demand peak. This can be seen by examining the net demand peak P t,d,net , for which the impact of PV is very limited. The same behavior is demonstrated for the maximum transformer apparent power load S max . These observations are consistent with the analysis in the previous section. On the contrary,

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backfeeding quantities, E bf , E t,bf and P t,bf , primarily depend on the number of buildings and percentage of PV, with only small impact of heat pumps. This is probably because backfeeding occurs mostly in the summer, where heat pumps play a very limited role. For all transformer-level indicators, i.e. the first six in Figure 4.3, we may notice no correlation with parameters related to grid components and configuration (average distance between consumers l avg , cable type Ca, transformer rated capacity S n and reference voltage Uref ). All indicators are computed based on the power flow in the transformer, which results from grid simulation. However, even though energy losses in the main feeder have been considered, the remainder of the island was represented by a dummy load. As a result, the transformer load is not significantly influenced by grid components in the results. Unlike the above indicators, for the maximum feeder current I max and the various 95%w voltage indicators (Umin , Umin and Umax ), grid characteristics play a more important role. I max depends mostly on the number of buildings, heat pump penetration, and the construction quality Q. The neighborhood type T and reference voltage Uref also have a smaller impact. For the voltage indicators, Uref has the most direct influence, especially for Umax , as already seen in Figure 4.2. The maximum voltage also depends a lot on the number of buildings and PV in the feeder, N and r PV respectively. The cable type and length are furthermore influential, while loads in the remainder of the island also have an impact, as they alter the voltage at the transformer. The transformer rated capacity S n also reduces voltage extremes, both 95%w overvoltage and undervoltage. Umin and Umin are similarly worsened by longer and weaker cables and transformers, seen as negative correlations, since the voltage values are lowered. In this case, PV has little to no influence, while neighborhood type and construction quality are more important, since they influence the heating demand. This section has identified the most influential grid parameters for several indicators, confirming the expected relationships. Overall, the feeder and island size naturally were the most important, with heat pump penetration levels driving demand indicators, and PV penetration driving backfeeding quantities. The neighborhood type and construction quality in the feeder were somewhat important as well, mainly for demand-related indicators. Grid characteristics were influential for voltage and current indicators, but not for transformer-level consumption and load quantities, primarily because energy losses were not modeled in the entire island. In absolute values, the linear correlation coefficients obtained reach very high levels for the most influential parameters, namely the number of buildings and percentage of heat pumps or PV, even above 0.9. Since Spearman’s rank coefficients (not shown here), were overall higher, it can be expected that linear models could be a good starting point for metamodeling.

Load profile analysis

4.3

|

85


In Section 4.1 we saw that overloading as well as voltage problems could occur for many grid cases at high penetration levels of heat pumps and PV systems. Section 4.2 furthermore showed which grid parameters are more influential for the various indicators. Under the studied scenario, heat pumps are expected to dominate network loading and cause the largest concerns, while renewable generation with PV doesn’t appear to alleviate the problems. To gain more insight into the mechanisms producing the observed indicators, this section analyzes simulation results, in particular electricity load and generation profiles. In Figure 4.4, annual load-duration curves are shown for three example cases of rural feeders with 30 buildings. The depicted cases differ in construction quality and penetration levels for heat pumps and PV. The various curves within each plot are created by successively adding time-series data for different load types, and sorting each series from higher to lower value. A time resolution of 30 min is used, so that the peaks coincide with those reported by the indicators. The annual peaks are discernible at duration zero, at the left end of each plot (also indicated with the horizontal dotted lines). It is, therefore, possible to observe how those change, starting from the base-load and adding heat pump loads (+HP), then back-up loads for domestic hot water (DHW) (+DHW BAK) and space heating (+HP BAK), and finally PV generation (+PV). At the far right, the larger backfeeding (negative) peak can be seen. These graphs depict individual replications, which might not represent the average of cases with same main grid parameters. Nevertheless, similar effects take place in all cases, allowing basic conclusions to be valid for the entire dataset. From Figure 4.4 we may first notice the base-load peak varies approximately between 35 and 45 kW, or roughly around 1.3 kW per building. This peak is much higher than synthetic profiles assuming 0.8 kW per building, because of the larger simultaneity in small LV grids, compared to larger collections of loads used to derive the synthetic profiles [213]. Addition of heat pumps (HP) to this base-load increases the peak in all cases, with larger impact at higher penetration level r HP , as suggest the comparison between the first and second example. Furthermore, comparing the second and third example, which represent a new and old neighborhood respectively, it becomes clear that less insulated dwellings cause larger peaks as well. From the additional peak caused by heat pumps, a large part can be attributed to the back-up electric elements used to boost the heat pump output during periods with ambient temperature below 0 °C (HP BAK). These are activated when heating needs are already high, and generally for all buildings at similar times, when the ambient temperature is very low. Again, the effect is larger for less insulated dwellings. On the contrary, back-ups for DHW heating (DHW BAK) have only a very small or negligible impact on the annual peak load for most cases. These are activated mainly during the

86 |


120 + HP BAK +PV

100 + HP BAK +PV

Neighborhood load (kW)

80

+ DHW BAK + HP

+ HP+ DHW BAK 60

+ HP BAK +PV + DHW BAK + HP Base-load

40

Base-load

Base-load

20 0 -20 +PV

-40 -60 -80

rHP = 30 %, rPV = 40 % 2

100

200

rHP = 60 %, rPV = 80 %

2

Uavg = 0.34 W/m K 0

rHP = 60 %, rPV = 60 %+PV

2

Uavg = 0.33 W/m K 300

0

100

200

Uavg = 0.59 W/m K 300

0

100

200

+PV 300

Duration (days)

Figure 4.4: Example load duration curves for rural neighborhoods of N = 30 buildings and varying levels of heat pump and PV penetration, as well as construction quality. Each curve is individually ordered after addition of a supplementary load. Starting from the base-load, the heat pump load (HP), DHW immersion back-up (DHW BAK), HP instantaneous back-up heater (HP BAK) and PV generation are successively included. Peak values are indicated by the dotted horizontal lines. night, and at different hours and days for all buildings in the neighborhood. Both back-up elements contribute little to the total consumption, however they can have important impact on the peak demand, because of simultaneity within the neighborhood. Better control of the heat pumps individually, for instance by earlier start or higher set-back temperature, could alleviate some of the problems with auxiliary heating. Nevertheless, load simultaneity remains an issue at the low-voltage distribution level. Furthermore, even for high PV penetration levels, as in the third example, production does not contribute in reducing the demand peak (see +PV). This is because the latter usually occurs during a winter evening, where little PV production can be expected. To examine further the issue of load synchronization, factors of simultaneity can be evaluated. The factor of simultaneity k s is defined as the ratio of simultaneous peak demand in a system, to the sum of individual peak demands, for any type of load in a given period: ¡ Pn ¢ max i =load 1 loadi k s = Pnload (4.4) max(loadi ) i =1


| 87

where n load is the number of loads, equal to the number of households, number of heat pumps, or number of PVs, depending on the type of load investigated, and max denotes the peak taken from the load time series. The maximum possible value for k s is 1, when the peak of all loads coincide. Here, k s is calculated at the distribution island level, on an annual basis and for 30-min resolution time series. For the total load, which includes the base-load, all heat pump loads and PV generation, the 30-min moving average is taken after 5-min loads are summed together. Figure 4.5 demonstrates how k s depends on the number of downstream consumers, or, more precisely, the number of loads, which depends on the low-carbon technology (LCT) penetration levels. For the base-load, k s drops from an average of 0.27 for 10 households to 0.18 for 200 households, with relatively small variation. A different situation can be observed for heat pump loads, with higher and more variable factors. For the heat pump alone (HP), when 10 heat pumps are present, k s reaches on average 0.62 and 0.87 for urban and rural grids respectively. For 100 heat pumps, these values drop to 0.52 and 0.63 respectively, much higher than for the base-load. This is because, certain times of the day, specifically the morning and evening, are common for the different heating patterns in residential buildings. The absence of buffer storage tanks or high set-back temperatures results in high simultaneity for these loads. Similar but lower are the simultaneity factors for the back-up elements activated at extreme cold conditions (HP BAK). For both of these loads, simultaneity depends significantly on the neighborhood type, as shown in Figure 4.5, but also on the construction quality. Rural and old neighborhoods tend to have higher simultaneity, because of their larger heating demand, which often also results in more frequent operation of the heat pump and back-up. These differences explain the larger variation observed for these loads. On the contrary, back-ups for DHW show very low simultaneity (DHW BAK), independent of building types or quality, since it only depends on the occupants. These back-up elements were used to heat up the hot water storage tank once a week for hygienic reasons, and to assist the heat pump in case of high demand. The latter operation was only occasionally necessary, while the assumed schedules for disinfection were diversified, resulting in low simultaneity. For PV generation, simultaneity is very high, stabilizing around 0.96. This result was expected, since the same weather was assumed for all buildings, and no shading was implemented. Consequently, the only diversification is a result of different orientation of the panels. When looking at the total electrical load, including all the previous, simultaneity is overall increased compared to the base-load, more variable and less stable. For rural grids, and also for older neighborhoods, k s increased more than for urban and new neighborhoods, as explained for heat pump loads before. The average increase is around 0.10 for urban feeders, growing from smaller grids to larger, and around 0.17 for rural feeders. However, more important are the extreme cases, which now surpass 0.6 for rural grids. Given that simultaneity factors for network sizing should

88 |


Total

1 0.8

Simultaneity ks

Base-load

HP

HP BAK

DHW BAK

PV

All data Rural Urban

0.6 0.4 0.2 0

0

100

200 0

100

200 0

50 100 150 0

50 100 150 0

50 100 150 0

50 100 150 200

Number of loads

Figure 4.5: Simultaneity factors ks for different types of loads. Total is the combination of all other loads. Gray dots represent all data points, while the lines denote the smooth fit for rural and urban islands separately. be on the safe size, revision of the current assumptions will be needed in presence of more heat pumps and other LCTs. A more detailed analysis with regard to the penetration levels could provide more specific values to serve that purpose. For the total load, some instabilities may be observed in the smoothed curves. These could be attributed to the fact that the total number of buildings in the grid (Nt ) is given on the x-axis, regardless of LCT penetration levels. This means that lower Nt values may actually contain grid cases with more heat pumps or PV than higher Nt values. As a result, inconsistencies exist between neighboring Nt values, given that the simulated dataset is limited, and each Nt does not cover the entire spectrum of heat pump and PV penetration levels. The present analysis of load profiles has shown that evolving towards electrified heating could lower electrical load diversity in residential neighborhoods, which should be taken into account for network design and management. In particular, back-up heaters operating during very cold days, could be responsible for a large portion of peak loads and resulting problems. To resolve these issues, improvements at the individual systems should be made. For instance, better control or additional thermal storage could reduce the size and operation time of auxiliary systems. Nevertheless, for higher penetration levels, action might be needed at the distribution grid level as well. Smart grid technologies, including smart metering and computer-based remote control, adapted to each feeder’s needs, could help improving grid stability utilizing the local flexibility. Demand side management (DSM) based solely on price signals may intensify distribution network problems instead, requiring feeder reinforcements [161]. Therefore, a form of smart management taking into account grid constraints is required [95, 153].

Cable and transformer overloading

4.4

|

89


Because of the change in loads and increased simultaneity demonstrated in the previous section, overloading of cables and transformers may occur for high heat pump or PV penetration levels. Section 4.1 already showed there are many cases with risk of transformer overloading. Since reoccurring overloading can cause deterioration of the grid component or even lead to failures, grid reinforcement or load management are necessary to maintain safe operation. Both options, however, require investments for new components, specialized equipment, installation and management. These are costs that should be included in the assessment of policies promoting LCTs. Of course, the rated capacity of a component determines whether it is overloaded, for a given load. The sensitivity analysis in Section 4.2 showed that this peak load, either the transformer maximum power S max or the feeder maximum current I max , depends primarily on the number of buildings and heat pump penetration, and less on the neighborhood type, construction quality and PV penetration. Based on those observations, this section presents in more detail the potential overloading situations, and how important grid parameters can influence the outcome. Note that both S max and I max represent the annual maximum values for normal operation, taken from 30-min averaged time series. As such, they provide an indication of potential overloading issues, allowing comparison between cases. For more detailed evaluation of transformer and cable overloading, the load profiles would be necessary, so that temperature rise can be calculated. For this work, the maximum indicators are considered sufficient to provide a relative assessment, as done in other similar studies [146, 158]. So far we have looked into indicators based on the entire simulated dataset as individual data points. Nevertheless, the proposed methodology gives the opportunity to analyze the results in a probabilistic manner, for each grid case in the main experimental design (Figure 3.2 on page 37). Indeed, for each grid case or design point, 20 replications are simulated, resulting in a distribution for each indicator. Figure 4.6 depicts as an example the histogram and cumulative probability of S max for one specific grid case. From this distribution, we may be interested in the mean, a percentile, e.g. p 90 , or the probability of exceeding the limit of 1 pu. The latter two are shown in Figure 4.6, and they can be obtained from the cumulative probability distribution function. In practice, the probability of a limit violation— which could also be a low limit—is calculated as the percentage of replications for which the indicator exceeded the limit, here for instance 70 %. This value could be of interest to network operators or policy planners, and is commonly reported in similar probabilistic studies [117, 118, 158]. Another potentially useful statistic of the distribution is the percentile, such as p 90 , which represents the value of S max

|


5

1

4.5

0.9

4

0.8

Counts

3.5

P(Smax >1 pu)

3 2.5

p90

2

0.7 0.6 0.5 0.4

1.5

0.3

1

0.2

0.5

0.1

0 0.85

0.9

0.95

1

1.05

1.1

Cummulative probability

90

0 1.15

Smax (pu)

Figure 4.6: Distribution of maximum transformer apparent power S max for an example feeder case based on 20 replications. Left y-axis: histogram counts. Right y-axis: cumulative probability. that has a 10 % probability to be exceeded for the given grid case. In this example p 90 is about 1.09 pu. The probability of exceeding the limit for transformer overloading P(S max > 1 pu), as defined in Figure 4.6, can be calculated for all design points in the dataset, namely n = 1296. We may then plot the proportion of grid cases that have this probability of overloading or smaller, for different subsets of the data, as illustrated in Figure 4.7. Here, the lines represent subsets with heat pump penetration levels within different ranges, for the entire distribution island or feeder respectively for S max or I max . This graph can be interpreted as follows: For a given probability of overloading on the x-axis, say P, the corresponding value on the y-axis determines the proportion of grid cases (design points) that have a probability of overloading equal or inferior to P. We may then see, for example, that for S max and heat pump penetration r HP,t ∈ [75, 100], 38 % of cases have zero probability of overloading, meaning that all 20 replications were below the limit of 1 pu. For the same subset, 100-46=54 % of cases have a probability equal to one, while 54-38=16 % of cases have a probability between zero and one. As expected, with higher penetration levels, the curves shift downwards, lowering the proportion of cases with no risk of overloading. The former depicts the proportion of grid cases with increasing probability of voltage violation, for the three voltage indicators and for different ranges of heat pump or PV penetration level in the feeder. For a given probability of voltage violation on the x-axis, say P, the corresponding value on the y-axis determines the proportion of

Proportion of grid cases with probability of overloading ≤P


|

91

1 0.8 0.6 0.4

rHP,t ϵ[0, 25) rHP,t ϵ[25, 50) rHP,t ϵ[50, 75) rHP,t ϵ[75, 100]

0.2


0 0

0.2

0.4

0.6

P(Smax >1 pu)

0.8

1 0

0.2

0.4

0.6

0.8

1

P(Imax >1 pu)

Figure 4.7: Proportion of grid cases (per subset) with a probability of transformer (left) or cable (right) overloading equal or inferior to P (given on the x -axis). Different lines denote subsets of the dataset based on heat pump penetration levels of the distribution island or feeder. grid cases (design points) that have a probability of violation equal or inferior to P. Table 4.1 summarizes the percentage of grid cases with a probability of overloading greater than zero, for different transformer rated capacities, and in total. Furthermore, for the same subsets, the percentage of overloaded transformers is given, when taking all individual simulations into account separately. The former percentage is always larger than the latter, since only one replication per design point needs to be overloaded to give a probability higher than zero. Note, however, that one might be willing to allow some probability of violation, for instance 5 %. Then fewer grid cases would be reported as problematic. As this table demonstrates, almost 68 % of grid cases with the 160 kVA transformer risk to be overloaded. This percentage drops to about 39 % for 250 kVA transformers, and 7 % for the 400 kVA ones. For the latter, mostly grids with large heat pump penetration levels had problems. These values suggest that the larger transformers could be sufficient even for important heat pump penetration levels. However, it should be kept in mind that replacement of distribution transformers can have significant costs [12, 146]. Furthermore, the studied cases have not considered other potentially important loads, such as electric vehicles, which could significantly lower the hosting capacity of these grids. Loading levels depend on several parameters, as already explained. With the help of contour plots it is possible to visualize the relationship between the most important ones and the transformer maximum load S max , as shown in Figure 4.8. For these graphs, the 90th percentile p 90 per grid case is used as starting point. Each contour

92 |


Table 4.1: Percentage of overloaded transformers, per rated capacity. Grid cases with probability of transformer overloading greater than zero (%) Overloaded transformers (%) Rural 160 kVA

250 kVA 38.9

400 kVA 6.9

All 37.8

61.9

32.6

5.0

33.2

Rural 400 kVA

Urban 400 kVA 0.8

2.5 2

60

0.4 1.5

40 1

0.2

Heat pump pen. rHP (%)

1

0.4

0.6

0.2

1 0.8

0.6

0.4

0.2

8 2. 6 2. 4 2. 2.2

1.8 1.6 1.4 1.2 1 0.8 0.6

80

Smax (pu) 3

1.2

2

100

160 kVA 67.6

20

0.5

0

0

0

0

0

10

30

50

70

90

110 13010

30

50

70

90

110 13010

50

100

150

200

Number of buildings N

Figure 4.8: Contour plots of the fitted 90th percentile of the transformer maximum load S max , on the distribution island heat pump penetration rate and number of buildings, for different neighborhood types and transformer rated capacities. plot depicts the surface fit of p 90 on the number of buildings and percentage of heat pumps in the distribution island, Nt and r HP,t respectively. The surface is a smooth fit performed with non-parametric local regression (LOWESS fit). Each contour plot in Figure 4.8 is the surface fit on a different subset of the data, split based on the neighborhood type and transformer rated capacity. Note that the left bottom of the plots may go to zero, but this is because of extrapolation, since there are no available data points in this region. Figure 4.8 confirms the already made observations that rural grids have larger overloading problems, which are reduced as the transformer rated capacity increases. We may notice that the small transformer of 160 kVA rated capacity would have problems even for small islands in rural grids, for instance starting at about 65 % penetration of heat pumps for 50 buildings. On the other hand, the 400 kVA transformer could support about 60 % penetration even for rural islands of 130 buildings and urban islands of 200 buildings, under the given assumptions. Mind that these surfaces are smoothed fits, therefore some cases lay also above the surface. Such kind of visual representations are useful to get a general view of the expected level of loading, and they are similar to the look-up tables proposed by NavarroEspinosa and Ochoa [158]. While these are easy to use and more intuitive, they only


1.5

0.8

0.9

0.6

0.7

0.5

0.4 0.3

1

0.3

60

93

Imax (pu)

Urban old

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4

1.1 1 0.9 0.8 0.7

80

0.6 0.5

0.4

0.3

40

0.5

0.2

20

0.

2

0.2


Rural old

Rural new

100

|

0

0

10

20

30

40

10

20

30

40

10

20

30

40


Figure 4.9: Contour plots of the fitted 90th percentile of the cable maximum current I max , on the feeder heat pump penetration rate and number of buildings, for different neighborhood types and building construction quality. Table 4.2: Percentage of overloaded feeders, per cable strength. Grid cases with probability of cable overloading greater than zero (%) Overloaded cables (%)

Moderate 12.7

Strong 8.3

All 10.5

5.3

4.0

4.7

give a rough approximation. The metamodels presented in the next chapters, can offer better accuracy and take into account all grid parameters. Unlike transformers, the simulated feeders have fewer overloading problems, only for 10.5 % of cases, as reported in Table 4.2. Figure 4.7 also showed that some probability of violation exists for heat pump penetration above 50 %. Evidently, overloading depends on the conductor rated current, suggesting the assumed moderate and strong cables can carry the demanded loads for most feeders, even at high heat pump penetration levels. Here, the difference between moderate and strong feeders is less pronounced, as the two cable types were close in terms of cross sections (see Section 3.4.2). A wider range of cable sections could provide more insights into the expected current-related problems, however, these should be selected carefully, to limit the required simulations while realistically representing potential options. More data on the current state of Belgian grids would be valuable for determining these cases. Same as for S max , Figure 4.9 presents some example contour plots for the 90th percentile of I max , for different neighborhood types and construction qualities. Figure 4.3 showed that the maximum current I max depends on these two grid

94

|


parameters. Indeed, we may see that again the rural and old feeders have the largest problems, with overloading starting from 50 % penetration of heat pumps for 45 buildings. Urban feeders, however, tend to remain below the limit even for 100 % penetration. Comparison between the rural new and old feeders indicates that construction quality, in particular increasing thermal insulation, could alleviate some of the grid constraints and delay grid reinforcements. It should be kept in mind, as for transformer loading, that these are thermal problems that depend on the duration of loads and other boundary conditions, and as such, the presented results are only indicative of the expected problems. This section has shown that for increasing heat pump penetration levels, transformers smaller than 400 kVA would have important overloading issues, if no measures are taken to reduce peak loads. Since other loads, such as electric vehicles, are not included in the analysis, grid reinforcement may be necessary to accommodate large numbers of heat pumps in the future. On the other hand, both strong and moderate cables appear to be sufficient for penetration levels at least up to 50 %. As explained in the next section, voltage constraints may be more important than cable overloading at the feeder level.

4.5

Voltage problems

Voltage problems are examined here, similar to the previous section’s analysis. As identified in Section 4.2, voltage indicators depend a lot on the reference transformer voltage Uref , but also on the feeder size, heat pump or PV penetration, average distance between consumers and the cable. Of the mentioned grid parameters, the discrete ones are used in Table 4.3 to show the percentage of violations per subset of the data. The neighborhood type is used to represent the large difference in average cable length. From Table 4.3, over the entire dataset, 33 and 36 % of grid cases have some probability of minimum voltage violation, while for the maximum limit only 19 % of cases have a risk. This observation again confirms that heat pumps would cause more voltage problems than PV, under the assumptions made in this work. We may, furthermore, notice the largest difference in terms of violations is found for Uref and the neighborhood type, thus the cable length. The construction quality Q makes a difference for the minimum voltage indicators, in particular the old neighborhoods compared to the other two. For the maximum voltage, construction quality makes no difference, as expected. Notice that a column is given for failed simulations, which are not included in the calculations of individual simulation percentages, but they were typically linked to important voltage drops. They are, furthermore, generally not accounted for in the calculation of the violation probabilities in the reported

Voltage problems

|

95

Table 4.3: Percentage of simulations with voltage violations, per reference transformer voltage Uref , cable type Ca, neighborhood type T and construction quality Q . Columns in gray give the percentage of grid cases with probability of voltage violation greater than zero, while the others report the percentage of individual simulations. 95%w Umin

Umin

Umax

Failed simulations

Uref

0.95 pu 1 pu 1.05 pu

39.2 18.6 9.9

53.6 27.5 18.6

52.1 17.4 7.1

66.5 28.2 14.0

1.0 5.7 23.6

3.9 13.5 39.3

1.6 0.3 0.8

Ca

Moderate Strong

26.3 19.1

38.4 28.2

28.9 22.4

40.9 31.9

11.4 8.5

20.8 16.5

1.5 0.3

T

Rural Urban

40.1 5.1

56.6 9.9

41.7 9.5

57.1 15.6

17.9 1.9

31.5 5.7

1.8 0.0

Q

Old Renovated New

26.4 20.8 20.8

37.9 31.6 30.5

29.6 24.7 22.8

41.1 35.6 32.7

9.2 10.0 10.6

18.2 19.4 18.5

1.4 0.7 0.6

22.7

33.3

25.7

36.4

9.9

18.7

0.9

Proportion of grid cases with probability of violation ≤P

All 1 0.8 0.6 0.4


0.2 0 0

0.2

0.4

0.6

0.8

P(Umin ≤0.85 pu)

rHP ϵ[0, 25) rHP ϵ[25, 50) rHP ϵ[50, 75) rHP ϵ[75, 100] 1 0

0.2

0.4

0.6

0.8

95%w≤0.9 pu) P(Umin

rPV ϵ[0, 25) rPV ϵ[25, 50) rPV ϵ[50, 75) rPV ϵ[75, 100] 1 0

0.2

0.4

0.6

0.8

1

P(Umax >1.1 pu)

Figure 4.10: Proportion of grid cases (per subset) with a probability of voltage violation equal or inferior to P (given on the x -axis). Different lines denote subsets of the dataset based on heat pump or PV penetration levels in the feeder. values. However, logistic regression metamodels for voltage violation prediction do take them into account in Section 6.3.6. The influence of heat pump and PV penetration on the voltage and the probability of violation can be determined from Figures 4.10 and 4.11. The former depicts the proportion of grid cases with increasing probability of voltage violation, for the three

96 |


Rural Uref =1.05 pu

100

Rural Uref = 0.95 pu

Urban Uref = 0.95 pu

Umin (pu)

0.5

5 0.6

0.9

5

0.5

0.7

60

0.8

0.6

0.7

0.7

5

5 0.8

0.6

5

40

0.8

0.9

0.7

5 0.7

0.8

5 0.8

5

0.9

0.8

5

0.9

0.9

20

1


1

80

0.6 0.5

0

0.4

10

20

30

40

10

20

30

40

10

20

30

40


Figure 4.11: Contour plots of the fitted 10th percentile of the minimum voltage Umin , on the feeder heat pump penetration rate and number of buildings, per neighborhood type and transformer nominal voltage Uref . voltage indicators and for different ranges of heat pump or PV penetration level in the feeder. For a given probability of voltage violation on the x-axis, say P, the corresponding value on the y-axis determines the proportion of grid cases (design points) that have a probability of violation equal or inferior to P. While there is a clear trend of downwards shift as the penetration levels increase (proportion of cases with zero probability of voltage violation drops), some lines intersect. The lines represent subsets of the dataset grouped based solely on one grid parameter, therefore, one-on-one comparisons should be made with caution. The reason is that the subgroups may contain somewhat different representation of other parameters, depending on the sampling, which may be more influential, such as the reference voltage Uref . For Umax , the differentiation between penetration levels is even less noticeable because of the latter. In Figure 4.11, contour plots are given for the 10th percentile of Umin , fitted on the feeder heat pump penetration level and number of buildings, for some example cases of rural and urban feeders with different Uref . Overall, as also clear from Table 4.3, the transformer reference voltage has a great impact, suggesting that solutions like on-load tap changers at the transformer could be critical in solving voltage problems. The middle plot of this figure, for Uref =0.95 pu, shows that the limit of 0.85 pu may be violated already from 25 buildings, even without heat pumps in the feeder. This is because heat pumps may be present in the rest of the distribution island, lowering even more the voltage at the transformer. This observation highlights the necessity to include other grid parameters in the analysis, such as done with metamodels, because two- or even three-dimensional representations may be misleading. The findings in this section demonstrate that wide integration of heat pumps in

The role of building properties

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97

existing Belgian LV grids would be problematic in terms of voltage levels. Rural and older neighborhoods are the first to be affected, because of the longer distances and larger, more heat-demanding dwellings. Overvoltage, caused by PV generation, was more limited for the studied scenarios (maximum 5 kW capacity), and only important when the reference transformer voltage was already above the nominal. As such, voltage rise in the MV at the same time, also due to renewable generation would aggravate the situation. To overcome voltage constraints, grid reinforcement, voltage regulation, energy storage or load management could be explored as solutions to allow important penetration of heat pumps and PV.

4.6


It was shown in Section 4.2 that building construction quality Q influences a majority of grid impact indicators. Since better insulated dwellings have lower heating demand, improvement of the buildings’ thermal insulation level could also allow larger penetration of heat pumps, and thus greater reduction in CO2 emissions. However, since Q is a categorical variable, not carrying very detailed information, this section investigates which particular building-related parameters can be linked to the grid indicators. These parameters could be incorporated in grid impact assessment tools, for instance, the metamodels described in Chapter 6, improving the accuracy when this information is available. Furthermore, they would allow to investigate the benefits of building renovation on the integration of LCTs. Because it is practically difficult to take into account characteristics of each and every building in the grid separately, aggregated parameters at the feeder or island level can be used instead. For this purpose, the parameters summarized in Table 4.4 were developed, which include average and total values, depending on the properties. Since construction properties of the buildings influence heating needs but not the base-load electricity demand, those were calculated only based on buildings equipped with a heat pump in each feeder or island. Similarly, parameters influencing PV production were only considered for buildings with a PV system. The chosen parameters include the floor area, air change rate, window-to-wall ratio and average heat transfer coefficient Uavg explained in Section 3.4.1. Additionally, the product U A is used to represent the overall thermal insulation, similar to Uavg , but taking into account the envelope heat loss area (A is not the floor area in this case). Similarly, g A is the product of solar heat gain coefficient and the building’s glazed area, giving an indication of solar heat gains. Both are summed over all exterior surfaces or windows of the building, respectively. The design heat load Q des is defined by Standard NBN EN 12831 [159] as “the heat supply needed under standard design conditions, in order to make sure that the required internal design

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Table 4.4: Overview of the considered neighborhood building-related parameters and the calculation method used to define them. Parameter

Symbol

Calculation method avg HP †

sum HP ‡

Design heat load*, kW (thermal) Floor area, m2

Q des

ΣQ des

A

ΣA

avg HP

sum HP

U A-value§, W/K

UA

ΣU A

avg HP

sum HP

g A-value§, m2

gA

Σg A

avg HP

sum HP

Air change rate, h-1

n ach

Window-to-wall ratio, -

wwr

avg HP

Uavg , W/m2 K

U avg

avg HP

PV rated capacity, kW

P 0,PV

avg HP

ΣP 0,PV

avg PV

sum PV

* Calculated based on NBN EN 12831 [159]. † Average among buildings equipped with a heat pump (or PV).

‡ Sum over buildings equipped with a heat pump (or PV). § A building’s U A- and g A-values are summed over all exterior surfaces or

total glazed area respectively.

temperature is obtained’’. In this work it also represents the heat pump rated capacity, since the systems were sized based on that. The heat pump capacity could be used when information on the heat load is not available, even though the two are not always equivalent. This is because often heat pumps are undersized, relying on auxiliary elements to satisfy occasional peaks in demand. Last, with regard to PV, the installed rated capacity is used as well. Similar to the analysis presented in Section 4.2, partial correlation coefficients are investigated for the building properties as well. Here each parameter is evaluated on its own, while controlling for grid parameters that were found to be influential for each indicator (Figure 4.3 on page 83). It is not the purpose to include all building parameters in a model to explain the outputs, but rather to find the ones with most potential, to be used in combination with the grid parameters in metamodels. In Figure 4.12 the Spearman partial correlation coefficients are shown between various indicators and the building properties of Table 4.4. For island-level indicators, i.e. the first six, the building properties were calculated over the entire island, and the total number of buildings, heat pumps and PV was used for the entire island. For other indicators, building properties were calculated only with respect to the feeder of interest, and control grid parameters were used for the feeder and rest of island, when appropriate. Here, instead of the heat pump and PV penetration levels, the number of heat pumps and PV are used instead. Since the building parameters were calculated only based on the buildings with those technologies, it is important to control for this interaction term, to avoid obtaining misleadingly high correlations.


Transformer-level

Spearman's partial correlation coefficient E t,bf

0.00

0.01

0.03

0.05

0.04

0.04

0.04

0.02

0.04

0.05

0.04 -0.16 -0.26

E t,d,net

0.02

0.05

0.22

0.21

0.34

0.25

0.41

0.10

0.26

0.28

0.43

0.03

0.03

Smax

0.03

0.03

0.17

0.11

0.29

0.18

0.37

0.07

0.27

0.19

0.38

0.03

0.08

Pt

0.01

0.02

0.12

0.07

0.25

0.12

0.31

0.05

0.24

0.12

0.32

0.01 -0.03

P t,d,net

0.04

0.07

0.32

0.20

0.51

0.35

0.69

0.15

0.52

0.36

0.69

0.04

P t,bf

0.01

0.02

0.03

0.03

0.01

0.03

0.01

0.01

0.01

0.03

0.01 -0.26 -0.37

I max

0.02

0.03

0.12

0.24

0.38

0.22

0.49

0.11

0.33

0.25

0.51

0.05

0.06

0.07

| 99

1.00 0.80 0.60 0.40

0.20

Indicators feeder-level

U min -0.01 -0.01 -0.07 -0.12 -0.18 -0.11 -0.21 -0.06 -0.19 -0.12 -0.23 -0.02 -0.03 -0.01 -0.01 -0.09 -0.13 -0.18 -0.14 -0.22 -0.08 -0.19 -0.14 -0.23 -0.02 -0.03 U 95%w min U max -0.01 -0.00

0.03

0.05

0.04

0.05

0.03

0.02

0.02

0.04

0.04

0.09

0.08

Ed

0.02

0.04

0.08

0.29

0.38

0.22

0.31

0.10

0.16

0.27

0.41

0.10

0.11

E d,net

0.02

0.04

0.09

0.28

0.38

0.22

0.35

0.10

0.19

0.27

0.43

0.05

0.06

Eg

0.02 -0.01 -0.00 -0.01 -0.02

0.00 -0.02

0.02 -0.01 -0.00 -0.02 -0.64 -0.81

E bf

0.01 -0.01

0.00

0.11

0.12

0.07

0.04

0.04

0.03

0.09

0.08 -0.26 -0.35

Pd

0.30

0.68

0.14

0.45

0.34

0.71

0.05

0.07

0.02 -0.02

0.01

0.00

0.03

0.72

0.90

gA

Qdes

Qdes P0,PV

0.02

0.07

0.18

0.32

0.52

P g -0.00

0.00

0.00

0.01

0.03 -0.00

wwr

nach

Uavg

A

A

UA

UA

gA

0.00 -0.20 -0.40 -0.60 -0.80 -1.00

P0,PV

Building-related parameters

Figure 4.12: Spearman’s rank partial correlation coefficients between indicators and different building-related parameters, having removed the effect of grid parameters. For the first six indicators, building parameters are calculated at the distribution island level, while for the others at the feeder level. For symbols, see Table 4.4 for building-related parameters and Table 3.1 on page 30 for indicators. In this case, Spearman’s rank correlations are preferred to the linear ones, since they were in general more conservative. From Figure 4.12 some general observations can be made. Overall, correlation coefficients are generally lower that the observed correlations for some of the grid parameters, in Figure 4.3 on page 83. This highlights the dominance of certain grid parameters, namely the number of buildings and penetration of LCTs. The largest correlation coefficients, up to 0.90, were obtained here for clearly PV-related indicators and the installed PV capacity P 0,PV . The electricity annual generation E g and peak P g are rather straightforwardly related with the PV systems capacity for given solar irradiance, and independent of occupant behavior or random effects. For demand-related indicators, for example P d and P t,d,net , building parameters reach considerable correlations up to 0.71. Of all heat pump-related parameters, the

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design heat load Q des showed the strongest correlations, followed by the U A-value, floor area A, average Uavg and g A-value. The average window-to-wall ratio wwr and air change rate n ach had almost no correlation with any indicator. Besides, we can notice a pronounced difference between average and total values for all properties. Since the coefficients in this analysis look only into first order effects, interactions with other grid parameters are not accounted for. As a result, the average values alone are not sufficient to explain variation in the indicators. Imagine, for example, 0.3 W/m2 K change in the average Uavg . The effect is much different if only one or 30 buildings are equipped with a heat pump. In a sense, the total values given in Figure 4.12 express the interaction between average values, the total number of buildings and the percentage of heat pump or PV. A sensitivity analysis method that takes into account interactions, such as the Sobol’ method, could provide more informative conclusions in that respect. Overall, this section showed that some building related parameters may have considerable correlation with the grid indicators. These relationships suggest that adaptation of the building design could facilitate heat pump integration in LV residential grids. Furthermore, the building parameters could be used to improve statistical modeling of grid impact indicators. For heat pump driven indicators, the design heat load Q des had the strongest correlation. However, also the total floor area, which may be more easily obtainable, had also a considerable impact.

4.7

Weather scenario analysis and simulation period

Weather conditions not only influence the heating demand, but also the generation of electricity based on renewable energy, such as solar or wind energy. Furthermore, air-source heat pumps consume more when the ambient temperature is low, not only because of reduced efficiency, but also due to activation of auxiliary electric resistance heating elements. As peak demand and simultaneity of loads are the main drivers of grid overloading, the influence of different weather profiles on grid impact indicators needs to be evaluated. The impact of weather, and in particular of climate change, on building energy use has been widely researched [40]. Generation of synthetic weather profiles for sensitivity and uncertainty analysis in building simulation is a topic of growing interest [74, 184]. Though sector-level analyses have been performed to estimate weather impact on electricity demand [126, 198], implications on the grid infrastructure have received little attention [86]. Furthermore, grid impact analyses are performed for specific or typical years [146], or only for few critical days [158]. This section, therefore, investigates the sensitivity of grid impact indicators to the


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101

choice of weather data. Furthermore, an analysis is made to determine whether a shorter simulation period, for instance a week, can produce the same result as the annual simulation. High dependence on certain weather attributes, such as the ambient temperature, could allow to select a representative week in this way.

Weather scenarios Various types of typical weather years have been developed and used in building simulations. Rastogi [184] offers an overview of those, as well as a review of synthetic weather generators. Meteonorm is a commercial software that provides such synthetic typical weather data for various locations [190]. This software focuses on solar radiation and temperature, which influence heat pump loads and PV generation the most. Additionally, it offers options to produce more extreme and variable weather conditions, as well as future climate change scenarios. As such, it was selected for the generation of annual weather profiles for this analysis. More specifically, nine weather profiles were generated with Meteonorm v.7.1 for Uccle, a weather station in the center of Belgium, often taken as reference in country-level studies. Using the advanced settings of this software, data series with monthly or annual maxima/minima for temperature and irradiation were produced. Furthermore, two of the nine profiles represent climate change scenarios for 2050. Table 4.5 gives an overview of all scenarios and Figure 4.13 shows the monthly mean temperature and irradiation. For all non future profiles, the software uses historical data from periods 1991–2010 for radiation, and 2000–2009 for temperature. Other advanced settings were kept the same for all profiles, for instance the starting seed (except scenarios 2) and the various radiation models. Scenarios 1 and 2 are average years, with average monthly irradiation and temperature, and average hourly temperature extremes (standard option). Scenarios 2-1 and 2-2 are created to investigate the impact of different first random seeds. Scenario 3 differs from scenario 1 in the temperature model, which uses 10-year hourly extremes. The mean monthly temperature remains the same in both cases, but the minimum hourly temperature drops from around -6 °C for the average years (-5.7 °C in scenario 1) to -9.7 °C. To simulate rare climatic events, a cold year is generated, whose mean annual temperature has a probability of happening once in a decade (scenario 4). Similarly, a sunny year has mean irradiation exceeded only once in 10 years (scenario 5). For extreme rare events, scenario 6 combines an extremely cold winter with an extremely sunny summer, which both increase heating loads in the winter and PV production in the summer. Within this year, each winter month has its lowest mean temperature

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Table 4.5: Weather scenarios with their temperature and irradiation settings, generated by Meteonorm 7.1. Label 1-Avg 2-1-AvgSeed5 2-2-AvgSeed10 3-AvgTempExtr

Description Average year,starting seed=1 Average year, starting seed=5 Average year, starting seed=10 Average year with extreme temperature variations

4-Cold 5-Sunny 6-CoW-SunS

Cold year Sunny year Extreme cold winter and sunny summer

7-2050Avg 8-2050CoW-SunS

2050 average year 2050 extreme cold winter, sunny summer

GHI (kWh/(m2 day))

Tamb (°C)

15

10

4 3 2 1 0

4

1

2

0.5

0

Average Yearly maxima Winter: average, summer: monthly maxima Average Same as 6

5

0

GHI

Tamb

5

Irradiation Average Average Average Average

6-CoW-SunS 7-2050Avg 8-2050CoW-SunS

3-AvgTempExtr 4-Cold 5-Sunny 6

1-Avg 2-1-AvgSeed5 2-2-AvgSeed10

20

Temperature Average Average Average Average (10 year extreme hour variation) Yearly minima Average Winter: monthly minima, summer: average Average Same as 6

0

-0.5

-2 -4

-1 2

4

6

Month

8

10

12

2

4

6

8

10

12

Month

Figure 4.13: Top: Monthly mean ambient temperature and global horizontal irradiation for all scenarios. Bottom: difference compared to scenario 1-Avg.


103

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for a decade, while each summer month has its highest mean irradiation for a decade. Such a year combines extreme scenarios for every month, making it improbable to occur as a whole. Future scenarios 7 and 8 are equivalent to scenarios 1 and 6, but for 2050. Among three options for future projections, the A1B scenario of the Intergovernmental Panel on Climate Change (IPCC) was chosen, because it showed larger deviations from scenario 1 in annual mean temperature and irradiation, but also in monthly means—warmer winter and less sunny summer. For all weather scenarios, 10 replications of the feeder cases described in Section 3.3 were simulated for the entire year. Based on the analysis in Section 3.5.3, 10 replications are considered sufficient for the present comparative investigation.

Impact of weather scenario Figure 4.14 shows the relative deviations in the transformer annual net demand E t,d,net and backfeeding E t,bf , for the different weather scenarios compared to the average scenario 1. Significant impact can be seen on both indicators for the extreme years. For the two extreme scenarios 6 and 8, the island annual net demand increased between 1 and 13 % compared to the average year, for the majority of cases (from p 5 to p 95 ). The increase of course depends on the percentage of heat pumps in the grid. We may also notice that for some cases in scenario 6 the annual net demand decreased, as a result of sunnier summer months. An equivalent increase

2-1-AvgSeed5 2-2-AvgSeed10 3-AvgTempExtr 4-Cold 5-Sunny 6-CoW-SunS 7-2050Avg 8-2050CoW-SunS -5

0

5

10

Et,d,net (%)

15

20

-20

0

20

40

60

Et,bf (MWh)

Figure 4.14: Change in annual energy demand and backfeeding compared to weather scenario 1-Avg. The central rectangle of the box-plots spans from percentile p 25 to p 75 , whiskers extend to p 5 and p 95 , the middle bar marks the median, and × mark the minimum and maximum values.

104 |


in backfeeding can be seen for those scenarios as well. Note that the deviations for E t,bf are given in MWh since backfeeding was close to zero for many cases, leading to large percentage deviations. For the average scenario 1, 10 % of cases had no backfeeding, the median was 45 MWh, p 90 was 150 MWh, and the maximum was 350 MWh. For the cold year, scenario 4, the net demand only increased, but only up to 5 %. For the sunny year, scenario 5, the net demand decreased by about 2.5 %, as a result of higher PV production. The future average scenario 7 was generally less sunny, with a decrease in backfeeding and small increase in net demand. For other scenarios, the differences are smaller than 2 % and 5 MWh for net demand and backfeeding respectively, indicating that annual quantities are little impacted by daily variations. To assess the impact of weather on quantities that depend on peak values and coincidence of loads, Figure 4.15 shows the change compared to scenario 1 for three overloading and three voltage indicators. In accordance with Figure 4.14, the extreme weather scenarios 6 and 8 have overall the largest impact. For 90 % of cases, included within the box-plot whiskers, the peak demand P t,d,net increased up to 0.4 pu. For the most extreme cases, the peak increased by 0.8 pu. It is no doubt that such an increase can lead to transformer overloading in many cases. In fact, for those two extreme scenarios, Figure 4.16 shows an increase in overloading of about 6, 8 and 3.5 percentage points for the small to large transformers respectively. These values concern the percentage of overloading for individual simulation results, and not per design point. The increase is somewhat limited, considering the large rise in peak load, because many transformers were already overloaded. The backfeeding peak is much less changed, because, as will be shown later, this peak depends a lot on the random coincidence of base-load demand with PV generation. Since transformer loading is in most cases determined by the demand peak, changes in the latter are more important. For the same two extreme scenarios, the feeder maximum current I max and the minimum voltage indicators are affected in a similar way as the peak demand, as they are very much related. I max generally increased up to 0.16 pu for most cases, with extremes around 0.7 pu. The percentage of overloaded feeders nearly doubled for moderate feeders, according to Figure 4.16. This figure also includes the percentage of failed simulations, to avoid misinterpretation of the displayed percentages. Closer analysis of those cases revealed they were mostly linked to very low voltage levels, suggesting a general overloading of the system. However, since no values were obtained for each indicator, they are reported separately. The minimum voltage for the extreme scenarios 6 and 8 dropped further another 0.04 pu for most cases, with extremes dropping as much as 0.22 pu for Umin . As demonstrated in Figure 4.17, violation increased by 2.3 and 4.3 percentage points for the global limit and weekly limit respectively. Failed simulations also increased


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105

2-1-AvgSeed5 2-2-AvgSeed10 3-AvgTempExtr 4-Cold 5-Sunny 6-CoW-SunS 7-2050Avg 8-2050CoW-SunS -0.3

-0.15

0

0.15

0.3

0.45

0.6

-0.1

-0.05

Pt,bf (pu)

0

0.05

0.1

0.05

0.1

Umax (pu)


-0.2

0

0.2

0.4

0.6

0.8

-0.1

-0.05

0 95%w

Umin

Pt,d,net (pu)

(pu)


-0.2

0

0.2

0.4

Imax (pu)

0.6

0.8

-0.2

-0.1

0

0.1

0.2

Umin (pu)

Figure 4.15: Change in overloading and voltage indicators (Table 3.1 on page 30) compared to weather scenario 1-Avg. See Figure 4.15 for explanation of the box-plots.

106 |


160 kVA 400 kVA

250 kVA Failed simulation

Strong Moderate

Failed simulation

1-Avg 2-1-AvgSeed5 2-2-AvgSeed10 3-AvgTempExtr 4-Cold 5-Sunny 6-CoW-SunS 7-2050Avg 8-2050CoW-SunS 0

10

20

30

40

50

60

Overloaded transformers (%)

70 0

2

4

6

8

Overloaded feeders (%)

Figure 4.16: Left: Percentage of overloaded transformers, per transformer size, for different weather scenarios. Right: Percentage of overloaded feeders, per conductor strength, for different weather scenarios. by around 0.6 percentage points, suggesting even more voltage problems. On the contrary, the maximum voltage was not affected for a majority of cases, with extremes up to 0.04 pu both positive and negative. The percentage of violations only marginally increased. Scenario 3, which has average temperature but extreme hourly variations, demonstrates similar results as the two extreme scenarios in terms of peak-related indicators. For P t,d,net , I max and Umin , the change is even a little larger. This indicates a considerable dependence of the annual peak load and voltage on temperature, 95%w and consequently, on heat pumps and their back-up electric elements. For Umin , the impact of daily extremes is smaller, as values below the weekly 5th percentile are not considered. A colder on average year, scenario 4, also increases demand peaks and lowers voltage levels, but to a lesser extent. The sunny year, scenario 5, had the reverse effect, with more impact on the backfeeding load. Interestingly, for non-extreme scenarios, such as scenarios 2, 5 and 7, both positive and negative changes can be observed for most indicators, which couldn’t be justified based on any grid parameter. This highlights how sensitive these indicators are to the random coincidence of loads. While the base electricity loads remain almost the same for all weather scenarios, small shifts in temperature cause different peaks in heating demand, and can thereby alter the voltage profiles. Another instance where randomness has significant impact can be seen in the large increase in (negative) backfeeding transformer peak load P t,bf for scenarios 2-2 and 7. Even though mean irradiation is maintained at similar levels as for scenario 1 (see Figure 4.13), a single extreme irradiance peak combined with a low-demand weekday caused P t,bf to increase for many transformers.


U 95% min i

βi i i x i3 +

q X q X i =1 j 6=i

βi i j x i2 x j +

q− q X2 q− X1 X i =1 j >i k> j

βi j k x i x j x k

(6.1)

Linear methods

|

159

with yˆ the predicted response and q the number of predictors x. For categorical predictors, such as the neighborhood type, cable and construction quality, terms of order higher than one are omitted, as they are not meaningful. To avoid overfitting, ridge regression is explored as regularization method [87]. Ridge regression modifies the cost function minimized by the ordinary least squares (OLS) method given in Equation 5.6 on page 136, namely the residual sum of squares. Ridge regression adds a term that penalizes the coefficients’ magnitude: X X ˆ 2 + λ β2 (y − y)

(6.2)

where λ is the regularization parameter, controlling the penalty strength. λ is an additional hyperparameter that needs to be tuned in the model selection procedure presented in Algorithm 6.1. As no prior knowledge exists on the optimal λ, 50 values logarithmically distributed between 10−4 and 104 are used. Combined with three different polynomial degrees, a total of J =150 different hyperparameter combinations are each time evaluated for LR metamodels. To train the various ridge LR models, the Matlab function ridge was used. This function automatically normalizes all predictors to have mean 0 and standard deviation 1, so that the coefficients have the same weight in the estimation procedure. For the evaluation of regression metamodels, the accuracy measures described in Section 5.6 are used, evaluated on the validation and test set. More specifically, these include the mean squared error (MSE), root mean squared error (RMSE), maximum absolute error (MAE) and coefficient of determination R 2 , as defined in Equations 5.1, 5.2 and 5.3 on page 133. R 2 measures the amount of variation in the response explained by the model, with perfect predictions yielding 1. On the contrary, the RMSE and MAE are measured on the response’s scale and should be as low as possible. MAE is used here to evaluate the local approximation in the distribution tail. While all these measures are analyzed in the following sections, for the automatic model selection procedure the MSE is used as performance measure. Even though prediction accuracy is the main objective of the metamodels, simplicity is an additional benefit. For this reason, it is useful to additionally compare the best choices among the three polynomial degrees. If performance is only marginally improved with a more complex model, the simple one would be preferred. On the contrary, a complex model offering only small improvement in the mean performance measure could be preferred, if it also works towards satisfying model assumptions. Such example will be shown in the next section. These additional checks are not part of the automated model selection and validation procedure. They are performed manually for one iteration of the procedure, to give an overview of the situation.

160 |

Metamodeling grid impact indicators: Application

Ridge regression, just like OLS regression, requires not only that a linear relationship exists and that the model is well specified, but also that the residuals are IID with zero mean and constant variance. Therefore, residual plots are also consulted to assess model adequacy.

6.3.2

Peak demand: analysis of ridge regression model

A first simple metamodel is built with ridge regression for the 90th percentile p 90 of the feeder peak demand for electricity P d . As a reminder, this value is calculated without grid simulation, simply by aggregating the demand profiles of all buildings, for the same cases as the general dataset. All 1296 design points are used, given that results are available for all replications. Since this indicator is not influenced by the grid topology or components, the only predictors used are the number of buildings in the feeder N , the percentage of heat pumps r HP , the neighborhood type T , and the general construction quality Q. Because of the few predictors, this model is more appropriate for model analysis and interpretation, as visualization is still feasible for 3rd degree polynomials. This section discusses model adequacy checking, explains the model selection procedure based on the ridge parameter λ comparing with simple regression without regularization, and finally offers some insights on model interpretation. The CV model selection procedure has chosen a 3rd degree polynomial based on minimization of the MSE criterion. Test results for the final ridge model are displayed in Figure 6.2. The left panel shows the errors are well distributed around zero and remain below 8 kW in all cases (MAE in Table 6.2). In terms of percentage error, the feeders with lower values of P d have the less accurate predictions. The largest test percentage error was 12 %. Nevertheless, the overall fit is rather good, as visually presented in the right panel of Figure 6.2. The RMSE, given in Table 6.2 is around 2 kW for both the CV and test results, while R 2 is very high, above 99 %. In general, this metamodel can be, therefore, considered as a good approximation of the probabilistic framework results for this indicator.

Table 6.2: Cross-validation (CV) and test performance of ridge regression model for p 90 of P d . CV average CV std. deviation Test

MSE 4.40 0.46 3.63

RMSE (kW) 2.10 0.11 1.90

MAE (kW) 7.96 1.13 6.54

R 2 (-) 0.997 0.001 0.997

Linear methods

8

250

Residuals Loess fit

Observed response

6

Residuals

4 2 0 -2 -4 -6

RMSE=1.90

-8 0

50

100

| 161

150

Predicted response

200

200 150 100 50 R2=0.997

0 250

0

50

100

150

200

250

Predicted response

Figure 6.2: Performance on test set of the final ridge model for p 90 of P d . Left: Test residuals. Right: Predicted vs. observed response. All axes in kW. Model adequacy While the resulting model was assessed satisfactory for prediction, model assumptions are evaluated here, by means of residual plots. Figure 6.3a and b show the training set residuals for the 2nd and 3rd order polynomials respectively, both for their best values of ridge parameter λ. We can see that for the 2nd degree polynomial residuals exhibit a curved pattern, indicating the model is not adequate and suggesting the need for higher order terms. Indeed, the residuals for the 3rd degree polynomial are approximately evenly distributed around zero for all levels of the predicted response, showing no significant problem of heteroskedasticity. The bottom plots of Figure 6.3 furthermore support an approximately normal distribution for the residuals, with zero mean. These observations confirm the validity of the 3rd order model and justify the additional complexity.

Model selection For the chosen polynomial degree, the top plot of Figure 6.4 shows the MSE for different values of λ in the model selection CV procedure. The λ giving the minimum average CV MSE is chosen for the final model, namely λ = 0.0215. We may notice that no significant improvement in MSE can be attributed to penalization, since the MSE remains rather stable for λ values below 0.1, and then only starts to increase. The bottom plot of Figure 6.4 furthermore shows the evolution of regression coefficients for the same λ values, fitted on the entire dataset D. We see that the coefficients remain fairly stable for the region of low MSE, and only shrink towards zero for

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Figure 6.3: Residual analysis for ridge regression for p 90 of P d . Top: residuals against predicted response for the 2nd (a) and 3rd degree (b) polynomial. Bottom: for the 3rd degree polynomial residuals, normal probability plot (c) and histogram (d). All are based on the training set D . Residuals and the response are in kW. λ > 0.1. Based on this figure it may be deduced that penalization didn’t significantly improve the predictive performance of the model. To further estimate the impact of λ on the regression coefficients, Table 6.3 gives a comparison between the ridge estimated coefficients with the best λ, and the equivalent un-regularized LR coefficients. For the latter, also the 95 % CI for the coefficients is provided. Since residuals were close to normal, the CI is computed based on the coefficient standard errors of the LR model. Here again we notice the ridge coefficients are very close to the LR coefficients, well within the CI. As a result, the un-regularized LR model is used in the following to analyze the regression coefficients and the relation of predictors and the response.

Linear methods

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Figure 6.4: Ridge regression for p 90 of P d . Top: Cross-validation MSE against the ridge parameter λ for the 3rd degree polynomial. Bottom: Ridge trance plot, displaying the standardized regression coefficients against λ. Different line styles denote terms of the polynomial of different degree. First order terms and others with high coefficients are furthermore indicated on the graph.

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Table 6.3: Comparison of regression coefficients for ridge and linear regression (LR) for 3rd degree polynomial models for p 90 of P d . Both models are fit on dataset D , with the same predictors. For the ridge regression, λ = 0.0215. The first column gives the polynomial terms, where “:” denotes interaction. Confidence intervals (CIs) are built from the LR resulting standard errors. CIs containing zero are marked in bold. Ridge coeff. Intercept TR Q old Q ren N r HP TR : Q old TR : Q ren TR : N TR : r HP Q old : N Q ren : N Q old : r HP Q ren : r HP N : r HP N2 r2 HP

TR : Q old : N TR : Q ren : N TR : Q old : r HP TR : Q ren : r HP TR : N : r HP TR : N 2 2 TR : r HP Q old : N : r HP Q ren : N : r HP Q old : N 2 Q ren : N 2 2 Q old : r HP 2 Q ren : r HP r HP : N 2 2 N : r HP 3 N r3 HP

LR coeff. 3.796 5.836 3.959 3.205 1.409 -0.101 -5.920 -2.829 -0.380 -0.103 -0.156 -0.117 -0.090 -0.084 0.0103 -0.012 0.0040 0.223 0.125 0.126 0.076 0.0150 0.0041 3.56e-04 0.0144 0.0065 -2.95e-04 8.36e-04 3.19e-04 5.38e-04 -1.21e-05 4.14e-05 1.07e-04 -2.70e-05

95% CI of LR coeff. 3.438 0.097 5.948 3.615 4.071 1.359 3.322 0.559 1.453 1.128 -0.105 -0.177 -5.960 -7.788 -2.870 -4.766 -0.387 -0.527 -0.103 -0.143 -0.163 -0.330 -0.125 -0.299 -0.090 -0.137 -0.084 -0.133 0.0104 0.0076 -0.014 -0.025 0.0041 0.0030 0.224 0.171 0.126 0.072 0.126 0.107 0.076 0.056 0.0150 0.0142 0.0042 0.0018 3.55e-04 5.09e-05 0.0144 0.0135 0.0065 0.0056 -2.05e-04 -3.05e-03 9.43e-04 -2.01e-03 3.18e-04 -4.80e-05 5.38e-04 1.60e-04 -1.19e-05 -5.31e-05 4.08e-05 2.63e-05 1.24e-04 -7.68e-06 -2.74e-05 -3.33e-05

6.778 8.281 6.782 6.085 1.779 -0.033 -4.132 -0.973 -0.247 -0.063 0.004 0.049 -0.044 -0.036 0.0131 -0.002 0.0051 0.277 0.181 0.145 0.096 0.0157 0.0066 6.59e-04 0.0153 0.0075 2.64e-03 3.89e-03 6.85e-04 9.16e-04 2.94e-05 5.54e-05 2.55e-04 -2.15e-05

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Interpretation To assess the significance of different regression terms, the coefficient CIs may be used. When the CI contains zero, the regression term may be seen as insignificant, as the coefficient could be zero. In a linear model with uncorrelated predictors, it would be rather straightforward to determine the important predictors. However, this is not the case for a 3rd degree polynomial where the same predictor variables are present in many of the regression terms. In Table 6.3, CIs containing zero are highlighted in bold font for the LR model for P d . It is mostly high order terms that appear to be insignificant in this case. However, in general, it is possible that a main effect is insignificant while its interaction with other predictors is important. Figure 6.5 visually presents the impact of each predictor on the predicted response. These plots present slices of the fitted response surface, keeping other predictors constant. Here, an average case is highlighted as example, with dotted horizontal and vertical lines. It represents a rural feeder with 30 renovated buildings, and 50 % of them equipped with a heat pump. In each subplot, a different predictor is varied to show its impact. With this figure we can see the peak demand is lower for the urban feeder equivalent with the same number of buildings and heat pumps. Also, the peak drops from old buildings to renovated and new ones, as expected. The number of buildings has a rather linear impact, for fixed other predictors. This could be seen from the insignificant or very small N 2 and N 3 terms in Table 6.3. Last, the heat pump penetration level r HP also monotonically increases, with smaller gradient 95 % band prediction

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at lower values, where it may be expected the base-load electricity demand is still influential. These plots furthermore give an overview of the uncertainty in the model function estimation, as well as the uncertainty in prediction of new observations. Here the plots show a very narrow band for the function estimation. The prediction band is much wider, representing the region in which a new observation may fall with 95 % confidence. We have seen that regression coefficients may, to some extent, provide information on the influence of each regression term, but as high degree polynomials are used, these become less easy to interpret. Slice plots, such as in Figure 6.5 could provide a visual representation of each predictor’s contribution, but they also become less practical for models with many predictors. In general, linear regression models are rather transparent and offer good interpretability for low dimensional data and models. Furthermore, for this particular case, a good fit was obtained, with an R 2 above 0.995.

6.3.3

Annual net consumption: mean vs. all replications

The second indicator modeled with ridge regression is the transformer-level annual net electricity consumption E t,d,net . Compared to P d , this indicator has more predictors, summarized in Table 6.1 on page 153. This is due to dependence on the rest of the island as well as on PV generation. As shown in Section 4.2, this quantity was only marginally influenced by network characteristics such as the cable and transformer, therefore network predictors are not included. For this indicator, we are interested in the mean value per design point E t,d,net , for which different modeling approaches will be used.

Model for the mean per design point The first approach uses the mean value per design point as output of the ridge regression metamodel. Here, design points with fewer than 15 available replications were removed from the training and test data, resulting in 1279 and 423 points respectively. Same as for P d , the 2nd degree polynomial resulted in curvature in the residuals. Therefore the 3rd order model was selected, with an optimal λ equal to zero. The effect of the ridge penalty was very limited in all CV rounds, where λ values very close or equal to zero were selected. Figure 6.6 presents the final test residuals and model fit, showing a generally good performance in terms of model assumptions and the error magnitude. A small problem of heteroskedasticity exists, seen as the larger spread of residuals for large values of the response. This could potentially be improved with weighted regression

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Figure 6.6: Performance on test set of the final ridge model for E t,d,net . Left: Test residuals. Right: Predicted vs. observed response. All axes in MWh. Table 6.4: Cross-validation and test performance of ridge regression model for E t,d,net . CV average CV std. deviation Test

MSE 22.95 1.96 18.82

RMSE (MWh) 4.79 0.21 4.34

MAE (MWh) 18.42 2.92 24.41

R 2 (-) 0.999 0.000 0.999

or response transformation, as proposed in the next section for the minimum voltage. Here, as the problem is rather limited and the focus lies in comparison of models for the mean, no transformation is applied. The prediction error, i.e. the test residuals, remain below 15 MWh or 8 % for all except four cases. The error expressed as percentage of the response is generally larger for smaller islands, and it is below 4 % for 95 % of cases. Table 6.4 gives the CV and test results for this model. Based on CV results, the RMSE is about 4.8 MWh, while MAE is around 18 MWh. To put these results into perspective, the histogram of relative prediction errors is given in Figure 6.7 for the test set. The error at each design point is divided by the standard deviation of E t,d,net for the same point. This figure shows that for all but one case the prediction error for the mean is smaller than 1.5 standard deviation, and smaller than 0.6 of the standard deviation for 90 % of cases. We could say the prediction is therefore not far from the observed E t,d,net .

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Figure 6.7: Relative prediction error on test set of the final ridge model for E t,d,net . Each error is divided by the standard deviation of E t,d,net within the same design point. Model for individual replications Instead of modeling E t,d,net calculated for each design point, another possibility is to model E t,d,net for each individual replication separately. In this case, the regression model tries to predict each value separately, but can only predict the mean of all cases with the same predictor values, if no additional information is available to explain the deviation therefrom. A new ridge regression model was fit for E t,d,net per individual replication, using the same procedures as before. The dataset D in this case consists of 20 × 1296 points, and the test dataset E of 20 × 432. In the two datasets, 0.9 % and 0.4% respectively were failed simulations, which were not used for metamodeling. The left plot of Figure 6.8 shows the resulting test residuals of the best model. For this model, the 3rd degree polynomial with a λ value of 0.1468 gave the best MSE. Residuals show more important heteroskedasticity in this model, which should preferably be treated to obtain smaller standard errors and more efficient predictions [149]. For this example, however, the original model is retained. The residual errors for this model are much larger than for the mean, because they relate to an individual prediction, while no additional information was given. In terms of error relative to the standard deviation of the respective design point, such as defined for Figure 6.7, for 90 % of cases this lies below 1.7 standard deviations. For this model, predictions are the same for all replications of one design point, seen as clusters of points on the same vertical line in Figure 6.8. This common predicted value actually approximates the mean of that design point. To demonstrate

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Figure 6.8: Residuals on test set of the final ridge models for E t,d,net of individual replications. Left: Model with Q predictor for the construction quality. Right: Model with predictors Q des for feeder and rest of island buildings. All axes in MWh. Table 6.5: Cross-validation and test performance of ridge regression model for E t,d,net of individual replications. Additionally, the test performance for mean prediction based on individual results is given. CV average CV std. deviation Test Test: mean of design point

MSE 199.61 4.30 198.80 20.28

RMSE (MWh) 14.13 0.15 14.10 4.50

MAE (MWh) 62.85 4.88 63.63 25.66

R 2 (-) 0.994 0.000 0.994 0.999

the latter, the prediction for E t,d,net per design point is calculated based on the individual predictions. Table 6.5 gives the test results for the mean, together with the performance for individual prediction. Comparing with the respective test results of Table 6.4, it may be deduced that the model yields a similar result. In fact, the residual plot (not shown here) is very similar to the one in Figure 6.6. We may, therefore, conclude that a metamodel for the mean could use the mean value instead of individual replications without loss of accuracy, in case the same predictors are used. This approach is commonly taken with linear regression metamodels for stochastic simulations [43, 114].

Model for individual replications and more informative predictor We have seen that keeping the same predictors in a linear model using the mean or individual replications, approximately the same estimate for the mean is obtained. This is because the latter predicts the same mean value for all replications. However,

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additional information may be available to distinguish between those replications. In reality, the replications may represent neighborhoods with common main characteristics defined in this work as grid parameters (see Section 3.3). A model with this additional information could provide better estimates for individual replications, and potentially for the mean as well. Here, to include this additional information, the general construction quality predictor Q is replaced by two more specific predictors, namely Q des for the main feeder, and the same for the remaining island separately. These parameters represent the aggregate design heat load for the feeder or remainder island, taking into account only buildings with heat pumps. They were defined in Section 4.6, where the correlation of different neighborhood-level parameters with grid indicators was tested. The right plot in Figure 6.8 shows the resulting test residuals of the best model with λ = 0.1468. With this model, individual different predictions are made per replication. This results in overall better predictions for all replications, reducing the CV RMSE from 14.1 to 11.7 MWh and CV MAE from 63 to 52 MWh (Table 6.6). In terms of relative (to the standard deviation) error, this dropped from 1.7 to 1.45 standard deviations for 90 % of cases. Prediction for E t,d,net per design point has also slightly improved using this more informative model. As Table 6.6 reports, the test RMSE for the mean dropped from 4.3 MWh and 4.5 MWh for the previous models to 4.2 MWh. These results show, as expected, that additional information for individual cases can improve prediction at the individual level, as well as for aggregate statistics like the mean. With different techniques, such as quantile regression, similar results could potentially be obtained for other distribution statistics. In practice, limitations are posed when the information is difficult to obtain. Therefore, metamodels of different granularity could be useful to accommodate applications with different data availabilities.

Table 6.6: Cross-validation and test performance of ridge regression model for E t,d,net of individual replications, using Q des as predictor. Additionally, the test performance for mean prediction based on individual results is given. CV average CV std. deviation Test Test: mean of design point

MSE 137.20 2.25 137.02 17.52

RMSE (MWh) 11.71 0.10 11.71 4.19

MAE (MWh) 51.92 3.46 58.83 31.73

R 2 (-) 0.996 0.000 0.996 0.999

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Minimum voltage: response transformation

The last indicator modeled with ridge regression is the minimum voltage. Specifically, the mean value per design point U min is used as response. Other sample statistics will be assessed in following sections, but here the mean is chosen as potentially easier to model. Nevertheless, even for the mean, linear regression is not expected to perform well, given the highly skewed distribution of this variable (see Section 6.1). For this indicator, all predictors of Table 6.1 on page 153 are used, which of course include network characteristics. For this indicator, not only the 2nd degree polynomial, but also the 3rd degree one was found insufficient. As can be seen in Figure 6.9, while the RMSE is low, at 2.2 V for the test set, there is a poor local approximation at the distribution tail. The discernible curved pattern and higher variance in the low voltage region indicate this model is not adequate. For such problems, response transformations may improve the model, as explained in the following. 15

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Figure 6.9: Performance on test set of the final ridge model for U min . Left: Test residuals. Right: Predicted vs. observed response. All axes in V.

Logarithmic transformation of the response In cases with non-linearities, residual non-normality or heteroskedasticity, transformations may improve the performance of linear regression models [114, 149]. The logarithmic transformation, in particular, helps to both normalize the response and stabilize its variance. In this case, since the U min distribution was left-skewed (Figure 6.1 on page 154), reflection is needed first. Therefore, U min is subtracted from the reference voltage Uref for each design point. Log-transformation of the difference Uref −U min results in an approximately normal distribution, as shown in Figure 6.10.

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Figure 6.10: Distribution of the log-transformed difference Uref −U min , with a normal distribution fit, based on all 1296 design points. Unfortunately, interpretation of coefficients for transformed models becomes more difficult. Furthermore, special care is needed to predict in the original response scale, because simple exponentiation gives an estimate of the median, instead of the mean [149]. Here, back-transformation is done using Duan’s smearing estimate [46], which gives the following prediction in the original scale: n ¡ ¢ 1X = U − exp( y) ˆ × U exp ln(Uref,i −U min,i ) − yˆi min ref n i =1

(6.3)

ˆ are multiplied by the where the log-linear model’s exponentiated predictions exp( y) average exponentiated residuals of the same model (n is the number of data points). Figure 6.11 gives the log-linear model’s residuals both back-transformed and in the model scale. To check model assumptions, the latter should be consulted. The right plot thus suggests important improvement in terms of heteroskedasticity. Nevertheless, on the right extremity, a dip is discernible, also reflected as an overshoot in the original scale errors. Comparing results of the linear model in Table 6.7 with those of the log-linear model in Table 6.8, we actually see both the RMSE and MAE are worse for the latter. Furthermore, the performance is much more volatile, seen by higher standard deviation among CV iterations for all measures of the log-linear model. These results suggest the log-linear model is still not appropriate for precisely modeling the mean in the low voltage region. However, the same model showed

Linear methods

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Figure 6.11: Test residuals of the final log-linear ridge regression model for U min . Left: Residuals back-transformed to the original scale, in V. Right: Residuals in model scale. Table 6.7: Cross-validation and test performance of ridge regression model for U min .

CV average CV std. deviation Test

MSE 8.25 1.16 4.74

RMSE (V) 2.87 0.20 2.18

MAE (V) 13.17 3.31 12.96

R 2 (-) 0.985 0.002 0.992

Table 6.8: Cross-validation and test performance of the log-linear ridge regression model for U min . The performance measures are given in the original scale. CV average CV std. deviation Test

MSE 12.22 4.56 5.76

RMSE (V) 3.44 0.65 2.40

MAE (V) 31.51 10.62 29.82

R 2 (-) 0.978 0.007 0.990

rather good performance where normal operation is expected, above the limit of 195.5 V. As such, it could be used to obtain specific values after a selection has been made to classify cases in more general regions. This sort of classification problem is explored in the next sections.

6.3.5

Logistic regression model

The previous section showed the linear regression model could not predict accurately the voltage level for each design point, especially in the very low voltage region. An

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interesting alternative is to look into the probability of voltage violation for a given design point. This probability can then be used to classify cases (design points) as problematic or not. Section 6.1 defined the response variables P viol and Zviol , which respectively denote the proportion of violating replications within one design point, and the classification of a design point as critical or not, depending on whether P viol is greater than zero. An appropriate model for this purpose is binomial logistic regression. It is a generalized linear model, used to predict the probability of success p of a binary p outcome. The logit function ln( 1−p ) links this probability to a linear predictor function, such as the expressions on the right-hand side of Equation 5.4 and Equation 5.5. Equivalent to the latter, and using the same symbols, a second degree polynomial function for logistic regression would be expressed as:

ln(

q q q− q X X X1 X pˆ ) = β0 + βi x i + βi i x i2 + βi j x i x j 1 − pˆ i =1 i =1 i =1 j >i

(6.4)

Logistic regression doesn’t make the distributional assumptions of LR, and is typically fit using maximum likelihood estimation [82]. To train logistic regression metamodels, the fitglm function of Matlab is used, with a logit link for the binomial distribution. With this function, the response variable used to train the model may be expressed in one of two ways: either as a binary variable, such as Zviol , or as a binomial proportion, such as P viol . In the first case, the model predicts the probability for each design point to be classified as critical. In the second case, the outcome is the probability for a new replication within the design point to result in voltage violation. Based on the latter, the former indicator can be computed as well. In the next sections, both approaches will be compared. For the logistic models, the same general work-flow presented in Algorithm 6.1 is followed separately for both models using P viol or Zviol . The inner CV loop in both cases calls the fitglm Matlab function for four different polynomial degrees: linear, linear with interaction terms, quadratic and 3rd degree. For this type of model, regularization was not used, because the available algorithm in Matlab, namely lassoglm, was very slow. Overfitting of the final model is avoided, since the model selection algorithm is based on the validation performance. Nevertheless, stepwise variable selection is also explored, which could improve performance while also avoiding overfitting, by selecting only important terms. Given that statistical soundness of stepwise variable selection has been criticized by some statisticians [82], in future work also other methods should be assessed. The same two approaches described above were repeated using stepwise regression instead of fitting the full models, to allow comparison of the final models. Again, the same four degrees of polynomial are assessed, but this time only representing

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the upper limit for the selection algorithm, implemented with the Matlab function stepwiseglm. This method successively adds or removes terms from the regression equation, based on some criterion calculated from the training data. Among other criteria, the Bayesian information criterion (BIC) is used here, because it tends to penalize larger models more [83]. Different criteria could be assessed as part of hyperparameter tuning, but here only one setting was explored. The performance measure used to compare and select models in the outer CV loop is the Brier score, as defined in the following. In the classification context, such as for the Zviol response variable, model performance is often assessed in terms of accuracy, which is the fraction of correctly classified observations. Relevant metrics are the recall or sensitivity, i.e. the fraction of identified true events (critical design points), and the precision, i.e. the fraction of predicted critical cases that are truly critical. One would seek to find an optimum trade-off between the two, because high recall means more detected critical cases, while high precision means less false detections. These metrics depend on the chosen cutoff value used to classify the resulting probabilities as critical or not. Since the above measures rely on the selection of a cutoff value, and because accuracy can be misleading in cases of imbalanced classes, additional measures are advised to assess predictive performance [82]. For instance, the Brier score is an error measure, similar to the MSE for regression, that can be used to evaluate the accuracy of probabilistic predictions. It takes values from 0 to 1, with 0 representing perfect prediction, and can be calculated with the following formula [25, 82]: Brier score =

n 1X

(y i − pˆi )2 n i =1

(6.5)

where n is the number of observations, pˆ the predicted probability, and y is the binary response. This score can furthermore be used when the response is a known probability, to assess accuracy of the prediction in a regression-like manner. In that case, y in Equation 6.5 is not the binary Zviol , but the actual known binomial proportions P viol . Since this score is appropriate for both approaches to model voltage violations, it is chosen as main performance measure for the logistic models. For model selection, Brier score is calculated using the same variable as for model training, either P viol or Zviol .

6.3.6

Voltage violation detection with logistic regression

This section trains logistic regression models to predict the probability of violating the lower voltage limit of 0.85 pu or 195.5 V. Here failed simulations are considered as cases with violation, therefore, all design points are used for training and testing. Four different models are compared, two based on the binary response variable

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Zviol , and two based on the binomial proportion response variable P viol . For both responses, one full model is trained, and one resulting from stepwise regression. In all cases, the same predictors are used as for the linear regression model for Umin , which are summarized in Table 6.1 on page 153. Furthermore, as explained in the previous section, four options for the polynomial degree are investigated for all models. This section aims both to assess the benefit of using stepwise variable selection, and to compare the performance and usefulness of modeling based on the two different responses. The best polynomial degrees selected by the four modeling approaches varied significantly. For the classification problem with Zviol as response, for all 25 CV iterations the full-model approach chose models with only linear effects. The stepwise selection also included few interaction and 2nd order terms. Including all interaction or higher order terms in the full models resulted in overfitting and worse validation performance. On the other hand, the probability models using P viol as response obtained better results with larger models, since more information is contained in that variable. The full-model approach selected quadratic models most often, while stepwise selection found better performance when also allowing some 3rd order terms. Full 3rd degree polynomials also overfit the data with this model. To get an overview of the final performance of the four different modeling approaches, Table 6.9 gives the Brier score results, both based on calculation with Zviol and P viol in Equation 6.5. The general results may be judged very satisfactory, given that Brier score is below 8 % for all models, and below 1 % for the best one. Models built with one of the two response variables gave better Brier scores calculated based on the same variable. The difference is larger for the Brier score calculated based on P viol , for which the model using the binomial proportion showed much better performance, below 1 %. This is expected, since P viol carries more information than Zviol , and would generally be preferred, if available. Nevertheless, the model with Zviol can serve for comparison with other classification models. Stepwise model selection resulted in similar performance as the full model based on P viol . However, for the models based on Zviol , the addition of some higher order terms (with stepwise selection) gave conflicting outcomes for the two Brier score calculations, also with larger variability in the performance. The Brier scores gave an evaluation of model performance similar to the MSE in regression. However, these models can also be used as classification models, where a given feeder case must be classified as critical or not. Therefore, we are interested in the prediction for Zviol , regardless of the model type used. Since the outcome of a logistic regression model is a probability, classification is done based on a probability threshold, the cutoff. Depending on this cutoff, different accuracy, precision and recall are obtained. As mentioned before, high recall means more detected critical cases, while high precision means less false detections. Therefore the cutoff value may be changed to maximize the preferred one in different circumstances.

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Table 6.9: Cross-validation and test Brier scores based on calculation with Zviol and P viol , for the four different modeling approaches.

Brier score based on Zviol


Brier score based on P viol


Model based on Zviol Stepwise Full 0.036 0.040 0.013 0.008 0.031 0.047 0.065 0.013 0.062

Model based on P viol Stepwise Full 0.077 0.077 0.013 0.014 0.074 0.071

0.057 0.007 0.058

0.006 0.002 0.005

0.009 0.003 0.004

1 0.95

Recall / Precision

0.9 0.85 0.8

Precision

0.75

Recall

0.7 0.65

Full Zviol Full Pviol Stepwise Zviol Stepwise Pviol

0.6 0.55 0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cutoff

Figure 6.12: Test precision and recall against cutoff, for the final selected model of all four classification modeling approaches. Figure 6.12 shows this trade-off between precision and recall for a range of cutoff values and the different modeling approaches. An important difference exists between the models based on Zviol and P viol . Since the binary response Zviol was defined as 1 for P viol > 0 and 0 otherwise, in order to predict Zviol from the P viol model, a threshold close to 0 should be used to obtain the best prediction. Best here is meant as the optimal trade-off between precision and recall. With higher cutoff values precision is improved, but with significant loss of recall. A somewhat different picture is presented for classification models based directly on Zviol . In this case, precision and recall meet close to 0.5, as observed from CV results. An average cutoff optimizing accuracy or some other criterion could be selected based on CV to be used in future predictions on unseen data.

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From Figure 6.12 we may again observe that stepwise selection yielded very similar results for P viol -based models, however classification performance improved for Zviol -based models. This would suggest that models with few interaction or quadratic terms give a better fit than the purely linear model. However full higher order models begin to overfit the data, as already mentioned. Therefore stepwise selection could be beneficial in this case. The results on Figure 6.12 show that rather high levels of prediction accuracy could be achieved. On top of the test performance, Table 6.10 gives the relevant values from CV. For the models based on Zviol and a cutoff of 0.5, CV recall is on average above 90 %, and precision above 92 %. This means that more than 90 % of critical feeders are identified, while at the same time less than 8 % of non-critical cases are falsely identified as critical. For the models based on P viol and cutoff of 0.05, even higher performance can be achieved, reaching an accuracy above 95 %. Overall, the logistic models performed well in identifying critical feeder cases with risk of lower voltage limit violations. The model using the proportion of replications with violation within a design point P viol was shown to give better predictions in terms of probabilities, as well as for classification. An appropriate probability threshold must be selected for the latter, one that reflects the definition of cases as critical. We furthermore saw that stepwise variable selection slightly improved performance, especially of the Zviol -based model. While not analyzed here, a potential additional advantage of stepwise selection could be the easier model interpretation, because only few higher order terms would be used instead of a full polynomial model. Given that much more training time is required for stepwise regression (see Figure 6.18 on page 191), it could be useful to first assess the potential benefits on few CV iterations before it is included in the metamodeling methodology.

Table 6.10: Cross-validation and test performance for the four different modeling approaches. Cutoff values of 0.5 and 0.05 are used to calculate performance measures for the models based on Zviol and P viol respectively. Model based on Zviol Stepwise Full 0.952 0.944 0.017 0.012 0.963 0.933

Model based on P viol Stepwise Full 0.963 0.953 0.013 0.009 0.958 0.958

Accuracy


Precision


0.926 0.026 0.939

0.924 0.025 0.897

0.957 0.026 0.944

0.947 0.020 0.938

Recall


0.931 0.034 0.952

0.908 0.031 0.903

0.932 0.027 0.931

0.910 0.032 0.938

Neural networks

6.4

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179

Neural networks

Neural network (NN) models are explored here for minimum voltage indicators. Since linear regression was found insufficient to model accurately Umin , an attempt is made to obtain better predictive performance with NN. Except for the mean response, a model is also built for the 10th percentile p 10 . Furthermore, the influence of available replications and design points on the performance is investigated. Last, a classification NN is built for the detection of critical feeder cases in terms of minimum voltage violations, with the purpose to compare its performance with that of the logistic model.

6.4.1

Neural network models

Neural networks, and in particular feed-forward NNs, are frequently used in literature for supervised learning in both regression and classification problems, because they can easily model highly nonlinear relationships [1, 122, 185]. It is, therefore, expected that they will offer better accuracy in predicting indicators such as the continuous minimum voltage response Umin , compared to LR models. In this work, multilayer perceptrons are investigated, which represent the most common class of feed-forward NNs. The Matlab Neural Network Toolbox is used for metamodeling, both for function approximation and classification, with the fitnet and paternnet network types respectively. The general metamodeling strategy as described by Algorithm 6.1 on page 157 is followed for the NN metamodels as well. Specifics about the models and their setting are given in the following, with a summary presented in Table 6.11. Section 5.7 already described multilayer perceptrons. They consist in multiple layers, each containing several neurons, as illustrated in Figure 5.6 on page 137. For the input and output layer, the number of neurons is defined by the number of input features and response variables respectively. The number of hidden layers and their respective number of neurons may be chosen by the modeler. Determining the network structure is important in metamodeling [186], especially because the resulting accuracy may significantly change [163, 222]. Here, a single hidden layer is used, as often done in practice [30]. Most rules of thumb suggest a number of neurons for the hidden layer between the number of input and output neurons. Having 12 predictors (13 features after dummy coding), values between 2 and 16 neurons seem appropriate for the investigation. Each neuron in one layer has a weighted connection to every neuron in the following layer. Neurons furthermore contain an activation function, which is used to compute the neuron’s output based on the sum of weighted inputs it received from other

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Table 6.11: Settings for neural networks. Default values are used for settings not mentioned in this table. Regression

Hidden layers Neurons per layer Hidden activation function Output activation function Training function

Performance function Validation set ratio* Stopping rules

Classification 1 2 to 16 Hyperbolic tangent sigmoid Log-sigmoid Linear Softmax Levenberg-Marquardt Scaled Conjugate Gradient Bayesian Regularization Resilient Backpropagation Bayesian Regularization MSE MSE (Brier score) 20 % Minimum performance gradient: 10−4 Maximum number of epochs: 700 Maximum consecutive failures to improve validation performance* : 30 Maximum Levenberg’s damping factor† : 1010

* Except for Bayesian Regularization, where no validation set is used. † Only for Levenberg-Marquardt and Bayesian Regularization, which uses the former.

neurons. Common activation functions include the log-sigmoid (or logistic, or simply sigmoid), the hyperbolic tangent and the linear (or identity) functions. In this work, both the log-sigmoid and hyperbolic tangent are explored for the hidden layers, while for the output layer the default linear and softmax are used for regression and classification respectively (Table 6.11). Training a NN consists in finding the appropriate weights for the connections, that minimize the error in prediction. Several algorithms may be used for training, with different performance depending on the problem. Here, a subset of the available training functions of the NN Toolbox are used, shown in Table 6.11, based on a performance analysis available from MathWorks [144]. To limit computation time, and more importantly, to avoid overfitting, early stopping and/or regularization may be applied [186]. Stopping rules include setting a maximum number of epochs, or setting a threshold for some criterion, including limits for the validation performance. Values for such settings selected for this work are shown in Table 6.11. To limit computation time, only a single value is used for all. Regularization is only applied with the Bayesian regularization algorithm, which automatically selects the best regularization parameters. For the other training algorithms, only the early stopping rules are used to avoid overfitting. In the above, the different aspects of NN models were described, presenting each time the specific choices made for this work. Table 6.11 gives an overview of the

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| 181

different options investigated with the model selection CV procedure. The number of combinations J assessed in each CV iteration totals 60 and 90 for the regression and classification NNs respectively, including variation in the number of hidden neurons, activation functions and training algorithms. The performance measures used for NNs are the MSE for regression and the Brier score for classification, as introduced for the equivalent linear models in Section 6.3.1 and Section 6.3.5 respectively.

6.4.2

Minimum voltage: mean and p 10

Since linear regression was found insufficient to model accurately the entire range of the mean minimum voltage indicator U min , the same indicator is fit with a NN. Additionally, the 10th percentile p 10 of the distribution within each design point is modeled. While the latter could be more useful to assess potential problems, it is also expected to be more difficult to model. For training and testing, design points with less than 15 available replications have been removed. The same inputs are used as for the regression models, which were summarized in Table 6.1 on page 153. For both the mean and p 10 , the best results were obtained with the Bayesian regularization training algorithm, which was constantly selected as best in all CV iterations. The optimal number of neurons in the hidden layer varied, however. For the mean, 7 to 16 neurons were selected during the 25 different outer CV iterations, with 9 in the final model. For p 10 , CV chose between 6 and 14 neurons, with the final model having 10 neurons. Differences between the hyperbolic tangent and logsigmoid activation functions were generally small, with none of the two dominating the selection. The final models for both the mean and p 10 use the log-sigmoid function. Figure 6.13 shows the resulting model for the mean. Similar to ridge regression, larger errors can be seen in the left tail. However, in this case, residuals are better distributed, smaller in general and with smaller extreme values. Compare, for instance, the CV performance of the NN given in Table 6.12 with that of the linear

Table 6.12: Cross-validation and test performance of NN model for U min and p 10 of Umin .

U min


MSE 4.14 0.61 2.98

RMSE (V) 2.03 0.15 1.73

MAE (V) 10.51 1.66 9.35

R 2 (-) 0.993 0.001 0.995

p 10


14.53 2.58 8.27

3.80 0.34 2.88

20.39 4.41 17.34

0.981 0.004 0.989

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10

Observed response

Residuals

5

0

-5

-10 100

250

Residuals Loess fit

RMSE=1.73 150

200

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250

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R2=0.995 150

200

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Predicted response

Figure 6.13: Performance on test set of the final NN model for U min . Left: Test residuals. Right: Predicted vs. observed response. All axes in V. 20

Observed response

15

Residuals

10 5 0 -5 -10 -15 -20 100

250

Residuals Loess fit

RMSE=2.88 150

200

Predicted response

250

200

150

100 100

R2=0.989 150

200

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Predicted response

Figure 6.14: Performance on test set of the final NN model for p 10 of Umin . Left: Test residuals. Right: Predicted vs. observed response. All axes in V. and log-linear ridge models in Table 6.7 and Table 6.8 respectively. The RMSE has dropped to 2 V from 2.9 and 3.4 V respectively, while MAE is around 10.5 V for the NN, lower than 13 and 31 V for the ridge models respectively. We may, therefore, conclude the NN model improved the fit on U min considerably. Furthermore, in NN, unlike linear regression, no assumptions on the residual distribution are necessary to support model validity. In Figure 6.14 the performance for p 10 is presented. The same general trends exist as for U min , but with larger errors, especially few large extreme values. Similar outcome can be observed from the performance given in Table 6.12. Not only the RMSE and MAE are much higher, but also the explained variation R 2 is lower. The decrease

Neural networks

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in performance could be explained by the fact that a larger sample size is needed to approximate a sample quantile compared to the mean [220]. Therefore, the next section explores a metamodel for p 10 based on 50 replications instead of 20.

6.4.3

Minimum voltage: 20 vs. 50 replications

The previous section showed a discrepancy in performance for the mean and p 10 , which could be, to some extent, due to the limited number of available replications, namely 20. Therefore, this section investigates whether additional replications could improve the model for p 10 . The previous dataset is extended to have 50 replications for each of 1296 design points. Note that, design points with less than 15 available replications were removed in the previous model, 17 in total. With the addition of 30 new replications, only 5 design points still had less than 15 available results, meaning that all other simulations failed. For each design point, p 10 is re-calculated based on all available replications (now up to 50), and the new model is built. For testing, the original set of only 20 replications is reused, excluding cases with less than 15 available replications. For this model, the CV procedure chose a NN that consist of a hidden layer with 12 neurons and the log-sigmoid function, trained with Bayesian regularization. Table 6.13 gives the performance of this model, which can be compared with the equivalent obtained with 20 replications in Table 6.7. The CV RMSE has improved considerably from 3.8 to 3 V, with smaller variation among CV iterations, while MAE also decreased. The test performance, however, appears to have somewhat deteriorated, for instance RMSE increased from 2.9 to 3 V. The fact that test performance hasn’t increased might be because the test set still only includes up to 20 replications. To get an idea of the improved fit of this model, the final training set residuals are shown in Figure 6.15 for both NN models with 20 and 50 replications. We can see for 50 replications the residuals are more concentrated around zero, and with less extreme values. This figure also highlights the design points where less than 50 or 20 replications are available, showing clearly that failed simulations are linked to low voltage. Comparing the two models, the one with 50 replications has more available

Table 6.13: Cross-validation and test performance of NN model for p 10 of Umin based on 50 replications. CV average CV std. deviation Test

MSE 9.40 1.79 9.06

RMSE (V) 3.05 0.29 3.01

MAE (V) 16.61 3.04 20.69

R 2 (-) 0.989 0.002 0.988

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20

Residuals Loess fit Rep.15 avail. rep.)

Predicted violations (%) Umin model p 10 model 22.94 24.94 28.46 22.91 24.59 28.37

Actual violations (%) 22.67 29.09 23.07 28.84

observations with large test RMSE and inaccurate predicted percentage of violating design points respectively. As a conclusion, it would seem that in order to model percentiles, the approach with individual replications is not the best, given that the additional predictors didn’t offer sufficient information.

6.4.6

Voltage violation detection with neural network

A final NN model was trained for classification, in order to compare the performance with logistic regression presented in Section 6.3.6. In this case, the NN is a classifier that uses the class labels provided in the binary response variable Zviol to compute the probability of an observation to belong in one class or the other. Here there are two options for the class label, namely critical or not critical. Table 6.16 shows the resulting performance for this model, in comparison to the performance obtained from the best logistic regression models obtained with stepwise selection using Zviol and P viol as response (Section 6.3.6). Overall, the NN model and the logistic model based also on Zviol give very similar results in terms of test and CV performance, with only slightly better results obtained with the logistic regression model. The model based on P viol remains somewhat better than the other two. Compared to NN, logistic regression furthermore offers the possibility to interpret the regression coefficients. It also requires less training time in total, even with stepwise selection, considering that less hyperparameters need tuning (see Figure 6.18 on page 191). Consequently, for this type of problem and given dataset, the logistic model based on P viol is the best option.

Discussion

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189

Table 6.16: Comparison of cross-validation and test performance for the NN classifier and the logistic regression models trained with stepwise selection, from Section 6.3.6. Cutoff values of 0.5 and 0.05 are used to calculate performance measures for the models based on Zviol (including the NN model) and P viol respectively.

Accuracy


Logistic Zviol 0.952 0.017 0.963

Logistic P viol 0.963 0.013 0.958

NN 0.943 0.010 0.951

Precision


0.926 0.026 0.939

0.957 0.026 0.944

0.921 0.023 0.931

Recall


0.931 0.034 0.952

0.932 0.027 0.931

0.907 0.033 0.924

Brier score Zviol


0.036 0.013 0.031

0.077 0.013 0.074

0.042 0.007 0.038

Brier score P viol


0.065 0.013 0.062

0.006 0.002 0.005

0.056 0.007 0.053

6.5

Discussion

6.5.1

Linear models vs. Neural Networks

The previous sections developed simple linear models and NNs for few indicators. Ridge regression was used, as well as logistic regression, and several choices were analyzed for the NNs. The predictive performance of each model has been already analyzed in the relevant sections, with comparisons where appropriate. This section aims to make a broader comparison of the two major categories of metamodeling techniques, namely the linear models and NNs. Apart from accuracy, simplicity, interpretability, and computational efficiency are discussed. When it comes to prediction accuracy, Section 6.4.2 showed that a NN model for the mean minimum voltage U min performed better than the equivalent ridge regression model. The latter had also problems in the residual distribution, which violated basic model assumptions. As a result, also inference properties of the linear model would be unreliable. In this case, therefore, the NN provided a better solution. On the contrary, in Section 6.4.6, the NN performance for classification of feeder cases as critical was found almost equally good as that of the equivalent logistic model.

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Since training of the NN also required more time, the logistic model seems to be a better option. It must be highlighted again that reported accuracies are with respect to the simulated response, which could potentially differ significantly from the real response. In terms of simplicity, the linear models, including ridge regression and logistic regression, are easier to understand, both conceptually and mathematically. However, confusion may exist with regard to the underlying assumptions and consequences of their violation. If the goal is to use the model for inference, conclusions will be drawn about each predictor’s contribution and interactions. To be more certain about the validity of such conclusions, deeper understanding of regression analysis and the underlying assumptions is necessary. Neural networks, on the other hand, have more complex structure and fitting procedures. Nevertheless, since software packages are nowadays available for implementing NN models, a user only concerned with prediction might find them as easy to use as linear or logistic regression. As often mentioned, a main advantage of linear models compared to NN is their transparency and interpretability. Section 6.3.2 demonstrated an example of coefficient interpretation. However, as pointed out, inference becomes less straightforward when higher order terms and interaction effects are included, especially for models with many predictors. These are, of course included in an attempt to improve accuracy. For more concise and interpretable models, methods for automatic variable selection exist, but they may have some issues associated with them. Stepwise selection, for instance, has received criticism with regard to the statistical properties of the resulting estimates [82]. Lasso regression also has limitations with regard to grouped variables, such as dummy variables for a categorical predictor [83], which means more advanced and less available algorithms should be used. Despite these potential difficulties with linear models, NN cannot compete in this area, since they offer no transparency. A last and critical aspect for metamodeling is computational efficiency of the method. Figure 6.18 gives for different models the training time necessary for one iteration of the inner CV loop, as defined in Algorithm 6.1 on page 157. It is clear from this figure that NNs require in general much more time for training than ridge regression, despite the fact that more settings are used in ridge regression (J = 3 × 50 = 150). Especially for the model of Umin with all replications, training time increases dramatically because the dataset is 20 times larger, while also more predictors are used, compared to the annual demand E t,d,net . Even for the models per design point, considering that often many CV iterations are necessary, here R × K × K 0 = 125, these differences result in many hours against few minutes for the NN and ridge regression respectively. Logistic regression was also very fast, when only the full models were built. The stepwise selection algorithm increased training time dramatically, especially for the model with P viol , because more higher

Discussion

p90 Pd

0.04

mean Et,d,net

0.62

all rep. Et,d,net

|

191

Ridge: J=150 Logistic: J=4 NN: J varies

19

mean Umin

9.2

full Zviol

4.1

full Pviol

3.8

step Zviol

120

step Pviol

323

mean Umin

J=60

all rep.Umin

263

J=30

Zviol

2446

J=90 38 0

50

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300

350

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450

Training time (s)

Figure 6.18: Average training time required for one inner CV iteration for each of the different models used in this thesis. It consists in the total time needed to train all J different settings. Overhead time for other operations during the metamodeling procedure is not included. order terms were necessary. This method becomes much slower as the number of predictors and polynomial degree increase, in which case the total time might be comparable with NN, depending on the number of different settings evaluated. The number of training examples clearly plays a role, as explained already for the NN of individual replications. Additionally, the stopping rules in NNs can alter training time, while also the available information in the dataset can speed up convergence. Furthermore, the number of different combinations of settings naturally changes the total training time. This can be very high with NNs, where many options exist for the network architecture, training algorithms, etc. Obviously, the software implementation and hardware play a role in the recorded times. As a consequence, the numbers presented in Figure 6.18 should not be generalized in any way. The overall conclusion, however, remains that for linear models, training time can be significantly shorter. This also suggests they are the best models to start with, when a metamodeling task is undertaken.

6.5.2

Potential of metamodels as decision support tools

Overall, the metamodels developed in this chapter showed there is promising potential in metamodeling for grid impact assessment. Ridge regression for the peak feeder demand performed very well, with a coefficient of determination above 0.99, and an equivalent RMSE below 2.1 kW. For the mean transformer annual net

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consumption, even better fit could be achieved, with an RMSE below 4.3 MWh. Minimum voltage values were harder to model, especially for small percentiles, for instance here p 10 . For the latter indicator, the best NN model based on 50 replications could reduce the CV RMSE to 3 V, or 0.013 pu. However, the performance of these models declined in the region of very low voltage. Nevertheless, logistic regression models could discover about 93 % of critical feeder cases for minimum voltage violations. These results suggest metamodels have the potential to replace simulation frameworks for grid impact assessment rather accurately. Of course, the accuracy against the real impacts will depend on whether the simulation models and framework are sufficient. Unfortunately, data to perform such validation for disaggregated loads or network measurements were not available. As discussed already in Section 2.4.3, metamodels such as those developed in this chapter can substitute the complex and time consuming probabilistic framework in many applications. For instance, they could be used in policy support concerning the adoption of LCTs like heat pumps and PV, in early stage urban planning, or in the design and operation of distribution networks. To give an example of how such metamodels can be used, Figure 6.19 and Figure 6.20 present possible graphical representations of metamodel predictions. In Figure 6.19, the logistic regression model based on P viol (Section 6.3.6) was used to predict the probability of voltage violation for different feeder cases and increasing penetration of heat pumps r HP . Each line of any plot represents a slice of the response space, where all other inputs except r HP are fixed. With many inputs, this is necessary in order to visualize the results. For example, in the top left subplot, the reference voltage Uref is 0.95 pu, the neighborhood type is rural and the average cable length l avg is 20 m. The number of buildings in the feeder N is different for each line. Other inputs are fixed at the same values for all plots in this example, namely, weak cable, old neighborhood, 50 % PV, 250 kVA transformer, 50 buildings in the remainder of the island, with 50 % heat pumps and 50 % PV. With such plots, a distribution system operator (DSO) could identify, for a given network, the maximum acceptable penetration of heat pumps before a certain probability threshold for voltage violations is exceeded. For example, in the top left subplot, 5 % probability of voltage violation in a feeder of 15 buildings would be reached at 45 % penetration of heat pumps. Similar plots or calculations could be made for other parameters, for instance different transformers, looking to increase hosting capacity by performing grid upgrades. Figure 6.20 gives a similar example, but considering two varying inputs, here the number of buildings and heat pump penetration. These plots are similar to the ones obtained in Figure 4.11 on page 96, only based on the neural network predictions of p 10 of Umin for other fixed inputs, instead of a simple surface fit through all data, as was done for Figure 4.11. Here other inputs are fixed similarly to the explanation given for Figure 6.19. These plots could be used to visualize potential

Discussion

Uref =0.95 pu

Uref =1 pu

Uref =1.05 pu

0.8

Rural lavg = 20 m

Probability of violation

1

| 193

0.6 0.4 0.2

0.8

N = 45 N = 40 N = 35 N = 30 N = 25 N = 20 N = 15 N = 10

0.6 0.4 0.2

Urban lavg = 7.5 m

Probability of violation

0 1

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Figure 6.19: Probability of minimum voltage violation plots, based on prediction from logistic regression model, for selected feeder cases.

40

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0.85

60

0.5 0.55 0.6 5 0.6 7 0. 0.75 0.8

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p10 of Umin(pu) 1

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80 Num 15 20 (%) 60 ber 25 30 40 . r HP of b 35 n e 20 p uild 40 45 0 mp ing t pu sN Hea

0.5 0.4

Figure 6.20: Contour plot (left) and 3D surface plot (right) of predicted 10th percentile of minimum voltage with NN, per number of buildings and percentage of heat pumps, for example rural feeder cases with Uref =1 pu.

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interaction effects between two inputs. Figure 6.20 gives merely an example of such plot. However, similar to this 3D surface, the metamodel response can be optimized with respect to several inputs. One could, for instance, minimize the peak transformer load, optimizing the penetration of heat pumps and PV, or optimizing the transformer and cable capacities. Since the metamodel approximates the grid indicators computed from the probabilistic framework, any other investigation that would be based only on those indicators and the grid parameters can be performed with the metamodel, in a much faster way and with little data requirements. Of course, the appropriate indicators should have been extracted for the creation of metamodels, according to the needs of each application.

6.6

Metamodeling application: Conclusion

This chapter applied the developed metamodeling methodology to the dataset generated for Belgian residential low-voltage distribution grids with heat pumps and PVs. The aim was to demonstrate in practice the metamodeling process, with emphasis on the effect of different choices, and to also assess the performance of simple and more complex models for certain indicators. Specifically, linear regression and logistic regression models were trained, as well as neural network models. To serve the purpose of this analysis, predictors and response variables of increased complexity were first selected, which helped demonstrate the different aspects of metamodeling. The response variables included the feeder peak electricity demand P d , the island annual net electricity consumption E t,d,net , and variables related to the feeder minimum voltage Umin . From the latter, the proportion of replications with voltage violations was defined as variable P viol . Also a binary response Zviol was created, which classified feeder cases as critical or not based on the value of P viol . The latter two were used to model the probability of having minimum voltage violations within a certain type of feeder case (design point). The methodology used to train and validate all models, linear or neural network, for regression or classification, was based on the general guidelines provided in Chapter 5. An outer loop of repeated cross-validation served for model validation, while an inner cross-validation loop selected the best settings for the different models. These settings included the degree of polynomial and ridge parameter for linear regression, and the number of neurons, activation functions and training algorithm for the neural networks. The final models were evaluated based on their cross-validation performance, as well as on the predictions for a new test dataset.


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The first model assessed was ridge regression for the 90th percentile of P d . It was shown based on residual analysis that a 3rd degree polynomial was necessary to obtain a valid model. In addition to model adequacy checking, also the effect of the ridge parameter on model performance was analyzed, showing that penalization does not always improve the fit. Furthermore, interpretation of the model coefficients was discussed, and plots presented that help understand the influence of each predictor. Nevertheless, inference in this way is limited to few predictors and low polynomial degree. Next, a comparison was made of linear models for the mean response, specifically for E t,d,net . Models using the sample mean from each design point were compared to models using all available replications as separate training examples. When the latter included additional information about the buildings’ thermal requirements, the individual as well as mean response were better captured. Since in practice this additional information may not be available, metamodels of different granularity could be useful to accommodate applications with different data availability. A ridge regression model was then trained for the mean voltage per design point U min . Since the distribution of this response variable was highly skewed, a logarithmic transformation was applied to normalize it and try to improve the fit. These results suggested the log-linear model was still not appropriate for precisely modeling U min in the low voltage region. However, the same model showed rather good performance where normal operation is expected, above the limit of 195.5 V. As such, it could potentially be useful for modeling specific values for U min in that region, for a subset of cases that have been previously classified as not critical for low voltage violations. Another approach to assess the risk of voltage violations is to directly model the probability of a violation instead of the actual voltage level. With this probability, feeder cases can be furthermore classified as critical for violations or not. Logistic regression models were trained for this purpose. Different approaches were compared, with different response variables used for training, and with or without stepwise variable selection. The latter slightly improved the final performance, but it also required much more training time. The benefit of stepwise selection in terms of accuracy could be evaluated first for a small number of CV iterations. An additional advantage could be the easier model interpretation, when less terms are included. With regard to the response variable used for training, using the actual proportion of replications with violation P viol yielded better performance overall, since it contains more information than a simple label as critical or not (Zviol ). An average accuracy above 96 % could be obtained with this model, equivalent to about 93 % of critical cases detected, and less than 5 % false detections. The second part of this chapter investigated neural networks and the potential improvement in performance they may offer compared to linear models. Since

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ridge regression was found insufficient to model accurately the mean minimum voltage U min , the same indicator is fit with a neural network. This model improved the previous performance of ridge regression, reducing the RMSE from 2.9 to 2 V, while also limiting extreme errors. Additional to U min , the 10th percentile p 10 of the minimum voltage distribution was modeled. Errors for this response were fairly larger than for the mean, which could be partially caused by insufficient replications. To examine this, a model was trained with p 10 calculated based on 50 instead of 20 replications. This model indeed improved the CV performance, indicating that statistics at the tail of a distribution require more replications for the same accuracy. An additional problem occurs for the minimum voltage, for which simulations may fail at the low voltage region, reducing the available results. The model for p 10 of the minimum voltage with 20 replications was furthermore used to analyze the impact of sample size, that is, the number of design points. Depending on the required accuracy, the training dataset could be reduced. For instance the RMSE for this model increased from 3.8 V to 4.6 V if only 648 design points were used instead of 1296. However, this conclusion may differ for other indicators or model types, therefore, identification of the minimum required sample size should preferably be conducted for all metamodels used in any application. To analyze whether a model with all replications could improve the prediction for p 10 , a neural network was trained with all 20 replications per design point as training examples. It was shown that this model tended to overestimate Umin for replications below the mean of each design point, and respectively underestimate it for replications above. In other words, the variability within each design point couldn’t be captured sufficiently with the given set of predictors. As a result, while the mean value was well estimated, predictions for p 10 were consistently overestimated. Since there are many phenomena influencing the value of this minimum voltage, additional predictors could potentially improve this model. A final neural network model was built for classification of feeder cases with respect to their risk of voltage violations. This model was compared to the respective logistic models, in order to assess potential improvement. However, the performance was found to be similar for all measures, thus not justifying the additional computational effort. The last section of this chapter provided a more general comparison of linear models and neural networks. Apart from accuracy, simplicity, interpretability, and computational efficiency were discussed. In particular, the comparison of training time requirements suggest that neural networks should be used in cases when linear methods fail to provide accurate predictions. This is the case, for example, with minimum voltage, whose behavior becomes highly nonlinear in the low voltage region. Linear models, nevertheless, provide simpler and faster solutions for a first assessment, and additionally better transparency and interpretability. Overall,


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metamodeling was found to produce satisfactory accuracy with respect to the probabilistic framework’s simulation results. This last section also demonstrated some of the potential applications of grid impact metamodeling, making use of graphical representations that can be produced with the developed metamodels.

7

Conclusion

This thesis has developed a framework for probabilistic assessment and metamodeling of the impact residential low-carbon technologies (LCTs) have on lowvoltage (LV) distribution networks. This concluding chapter first explains the context for this research, and then summarizes the methodology and main findings. Recommendations for improvement and continuation of the developed framework are provided last.

7.1

Motivation and objectives

Dwellings have a role to play in the transition towards a more sustainable, lowcarbon and fossil fuel-free energy system. They have a large potential to reduce energy use by implementing energy efficiency measures, in particular with regard to heating demand in central and northern European countries. Thermal insulation and LCTs for heating have been promoted and often subsidized in light of the European commitments to combat climate change. Renewable energy sources are also high on the agenda of most countries, including distributed small generation by home owners. In Belgium, air-source heat pumps and rooftop photovoltaic systems (PVs) are technologies increasingly adopted in residential buildings. Despite the benefits of heat pumps and PV, there are challenges associated with their integration in the LV distribution grid. Overloading, reduced voltage quality, reverse power flow and increased grid losses are some of the potential issues from wide deployment of such technologies. To cope with these issues, grid upgrades, energy storage, load management or other measures would be required, resulting in additional integration cost. These issues are often ignored or simplistically considered in high-level assessments and in many urban-scale optimization approaches. Nevertheless, research has begun to focus on assessing the potential impacts of LCTs on the electricity system and the local distribution grid in particular. Estimation of these impacts requires load flow analysis, high resolution time-series

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inputs, and details of the network. However, because of the high computational burden, most studies in literature are restricted to the analysis of one or few specific networks under a limited number of scenarios. From this context, the need for a more systematic approach to grid impact assessment for LCTs becomes clear. Such approach should look into multiple different network cases, and consider inherent uncertainties related to buildings, their location, and their electricity demand and generation. In this way it can provide a probabilistic assessment of the impacts for a significant number of different cases. In order to be valuable for policy makers, distribution system operators (DSOs) or urban planners, the results of such assessments need to be easily obtained and communicated. This could be achieved with methods that allow to make approximate predictions while circumventing the tedious data gathering, modeling and simulation required by the complex probabilistic grid impact assessments. Metamodeling is one such method. Under these premises, the research objectives for this thesis were defined as the following: Research objectives d Describe and quantify the probabilistic impacts of low-carbon solutions in residential buildings on LV distribution grids. d Explore metamodeling as a method to allow a computationally inexpensive assessment of these LV grid impacts.

7.2

Summary of methodology and conclusions

Probabilistic impact assessment framework In order to fulfill the first objective set for this thesis, a framework was developed that calculates the probabilistic impacts of heat pumps and PV on a variety of distribution grid cases. The framework considers residential loads from heat pumps and PV, and LV distribution islands with radial three-phase feeders. There may be different types of impacts on those grids, as reviewed in Section 2.2. This work focused on issues occurring under normal operation, mainly component overloading and underand overvoltage. Other quantities, such as annual electricity consumption and backfeeding were also considered as impact indicators. The framework uses a Monte Carlo approach to compute probabilistic impacts for different network cases representing Belgian residential LV distribution islands. The grid cases are parameterized based on the number of connected buildings, the average distance between connections, the neighborhood type (either rural


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or urban), the feeder cable strength, the transformer rated capacity and reference voltage, as well as the general building construction quality. To take into account uncertainties in the loads, a building stock is created with buildings of varying properties and stochastic occupant behavior. Simulated load profiles from this stock are then randomly sampled into the grid cases in multiple Monte Carlo replications. Year-long load flow analysis calculates the grid impact indicators for each replication, resulting in probability distributions of the impact indicators, for each grid case and different LCT penetration levels. This framework can be used for probabilistic investigation of grid impacts. However, with the described approach, it also generates the input/output (I/O) combinations needed to train metamodels, which can then predict the impact indicators based on grid parameters and LCT penetration levels. Moreover, several studies were performed to determine the influence of methodological assumptions and choices, mostly made to reduce computation time, on the accuracy of resulting indicators. These studies analyzed simplification assumptions with regard to the distribution island model representation, temporal resolution requirements for the load profiles and power flow analysis, and the impact of different numbers of grid cases and Monte Carlo replications. As a trade-off exists between computation time and accuracy, the findings of these analyses, presented in Section 3.5, provide useful insights about the challenges of setting up such a methodology.

Impact of heat pumps and PVs on Belgian residential feeders Simulation results from the probabilistic framework were used to analyze the expected impacts of a wide deployment of heat pumps and PV in Belgian residential LV grids. From the examined grid cases, it was shown that rural and older neighborhoods would be affected first, because of the longer distances and larger, more heat-demanding dwellings. Some probability of undervoltage problems was found for 35 % of simulated feeder cases, while overvoltage was only probable for 20 % of cases. Probability for transformer overloading was found for 34 % of cases, a majority of which concerned the small 160 kVA-transformer. For all problems, the grid cases with more buildings and higher heat pump or PV penetration were the most affected. Indeed, sensitivity analysis on the grid parameters ranked the number of buildings in the grid as most influential for almost all indicators, with heat pump penetration levels driving demand indicators, and PV penetration driving backfeeding quantities. The neighborhood type, either rural or urban, and the buildings’ construction quality were also shown to have an impact on demand-related indicators, since larger and less insulated dwellings require heat pumps of greater capacity. Voltage and current indicators furthermore depend on grid characteristics. In fact, also other building-

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related parameters, such as the heat pump installed nominal capacity, were found to have important correlations with heat pump-driven indicators. This would suggest that such building parameters could be used to improve statistical modeling of grid impact indicators, while also allowing to assess scenarios of building retrofit. Overall, heat pumps and PV were found to cause important issues, starting from different penetration levels depending on the grid case. Since overloading can be solved by replacing the component, cable or transformer, with one of higher capacity, reinforcement could offer a solution to allow larger LCT penetration. Given the increasing ownership of electric vehicles, not considered in this work, grid reinforcements may be needed even at lower penetration than that estimated in this analysis. On the other hand, analysis of load profiles indicated that heat pumps would increase load simultaneity, in particular when equipped with electric auxiliary elements. Considering the flexibility of these thermostatically controlled loads, benefits could arise from the use of smart grid technologies for distribution network management. Other potential solutions that could be explored include energy storage and voltage regulation strategies. Apart from the main analysis, the effects of some boundary condition assumptions were further scrutinized. The first analysis focused on the effect of weather data selection. It was shown that for a conservative grid impact assessment, weather profiles capturing extreme temperatures should be preferred, since an average weather year could underestimate potential problems for networks with highly electrified space heating. The second study demonstrated that a more conservative power factor for heat pump loads would influence grid impact indicators significantly. As such, more research should be devoted to determine reactive power consumption of heat pumps, so that it can be included in grid impact studies. Finally, the last study looked into the building stock definition, specifically with regard to the assumed ratio of single-person households. Small changes in this ratio, affected the annual electricity consumption, but other grid impact indicators were minimally affected. Nevertheless, more research could be useful to analyze potential household clustering. Furthermore, spatially disaggregated consumption data could help verify the simulated base-load electricity demand.

Metamodeling methodology The second objective of this thesis was to explore metamodeling as a method to predict grid impacts in a simple and faster way, based on few input parameters. Such metamodels can then be used in high-level assessments, where multiple networks and cases need to be represented, but little time and data are available. Metamodeling literature focuses either on specific applications other than districtlevel assessments, or on developing algorithms for a particular technique. Therefore,


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a structured overview of available methods was compiled, describing challenges and solutions for each metamodeling step, relevant to grid impact metamodeling. This work considered metamodeling as part of the general probabilistic framework, since the simulation results discussed above are used as training data for the metamodels. Because of that, the probabilistic framework was adapted to cover more efficiently the input space, rather than to facilitate comparison of few specific cases. The metamodeling methodology begins with several preparatory steps, such as defining the goal, inputs and outputs of the metamodel, preparing the experimental design and choosing an appropriate metamodeling technique. While prediction is the main goal for metamodels of grid impact indicators aimed to be used as decision support tools, inference is also useful for problem understanding and determination of influential parameters. These latter tasks were performed directly on the detailed results in this case. In terms of output types, this thesis focused on time-invariant outputs, which are more appropriate for high-level probabilistic analysis. Given that a distribution of the output is obtained from the probabilistic framework, different approaches were discussed for metamodeling properties of this distribution. For the type of metamodels investigated here, inputs need to be time-invariant parameters that are easily obtainable for impact analysis, therefore, considerations are needed to derive neighborhood-level scalar inputs. These correspond to the grid parameters defining the grid cases to be evaluated by the framework. A space-filling experimental design can be used to select those cases, covering the input space more efficiently. This thesis furthermore briefly reviewed different metamodeling techniques for timeinvariant continuous outputs, appropriate for most of the grid indicators. Several techniques are generally considered when metamodeling, since all methods have advantages and disadvantages. Among other methods, this thesis further explored linear models, since these are simple and rather transparent, and neural networks as more complex and potentially more accurate alternative. Regardless of the chosen technique, the training and validation procedure consists in: (i) choosing a set of potential model settings or forms to investigate; (ii) training all models for those options; (iii) selecting the best settings based on the performance on validation data; (iv) evaluating the resulting metamodel on test data; and (v) repeating previous steps with new model types, if necessary based on previous outcome.

Metamodeling application In the last part of the thesis, linear models and neural networks (NNs) were used to generate metamodels for few grid impact indicators, demonstrating the metamodeling procedure in practice, highlighting the impact of different options and analyzing the performance of both models for this application.

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Ridge regression, a form of linear regression used to avoid overfitting, was employed to train metamodels of the feeder peak demand and to demonstrate how model selection, adequacy checking and interpretation are performed. Then the mean value for the annual net electricity consumption was modeled, based on a dataset using individual replications or the distribution mean per grid case, demonstrating that including additional parameters describing building thermal demand in individual replications could improve predictions. Since in practice this additional information may not be available, metamodels of different granularity could be useful to accommodate applications with different data availability. For both indicators, ridge models performed well, with an R 2 above 0.99. Ridge regression was, however, found insufficient to model minimum voltage indicators, especially in the low voltage region, despite using response transformation. Therefore, another approach was taken, based on logistic regression. The latter was used to model the probability of voltage violation for each grid case, classifying those cases as critical or not. Different models were investigated, with the best approach detecting about 93 % of feeder cases with probability of voltage violation, with less than 5 % false detections. Next, neural networks were explored for minimum voltage indicators, investigating whether better performance can be obtained compared to ridge regression. Indeed, the RMSE dropped from 2.9 to 2 V for the mean value of the indicator per grid case. The 10th percentile p 10 of the minimum voltage distribution was modeled as well, for which several modeling options were investigated. A low percentile such as p 10 is generally harder to model, but could be more useful for decision making. The reduced performance was improved when additional replications were used per grid case. Using all replications as training points to predict p 10 , however, didn’t bring any benefits, since this approach works better for modeling the mean response. Furthermore, it was shown that for this particular model, with half the sample size, i.e. the number of grid cases,the RMSE would increase to 4.6 V, compared to 3.8 V for the large dataset. For different models and indicators, however, the required sample size for a given accuracy may vary. Sequential addition of grid cases or adaptive sampling could be investigated in future work to reduce the number of simulations. Finally, classification with neural networks was also explored, but it yielded similar results to the logistic regression models, albeit with longer computation time. In general, the linear models performed well in terms of accuracy for the peak load and annual demand indicators, as well as in predicting the probability of violation with logistic regression. Furthermore, linear models are simpler, more transparent and generally require less training time than neural networks. The latter, more complex models, do not offer any interpretability, require much tuning and long training times. However, they may provide better accuracy for nonlinear problems, such as for the minimum voltage indicators. Given these observations, a recommendation would be to start with the simple models and only proceed to

Recommendations for improvements and further research

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more complex approaches when the required accuracy is not reached. Overall, the findings of this thesis suggest that metamodels could approximate the detailed results of the probabilistic framework with satisfactory accuracy. As such, grid impact metamodels could be used to facilitate decision making in various fields, as explained in Section 2.4.3. Applications could include, for instance, determining the feasibility or additional costs of policies promoting adoption of LCTs, supporting early-stage planning of district energy systems, or assisting DSOs in the design and operation of their distribution networks, by estimating future impacts without need for detailed simulations. It should be kept in mind that metamodels can only be as accurate as the original model they approximate. However, estimating the accuracy of probabilistic grid impact models is a challenge, especially when new technologies and future scenarios are investigated, due to lack of appropriate data.

7.3


The proposed probabilistic framework estimates impacts of LCTs on LV distribution grids, with the purpose to evaluate technical restrictions to their deployment. The impacts addressed with this framework relate to the steady state operation of the networks, at time-scales of few minutes. This thesis focused in particular on component overloading and voltage magnitude variations, however, other quantities, such as energy losses or voltage unbalance, can also be examined. Impacts on the overall electricity system are not explicitly quantified, but indicators such as the annual electricity consumption or peak load per distribution island could be employed towards that end, if calculated for a representative set of such islands. On the other hand, transient analysis is not part of the proposed framework. Furthermore, the framework does not analyze carbon emissions related to the operation of heat pumps, for instance, nor those resulting from the life cycles of any LCT. For a comprehensive assessment of LCTs, these issues should be also studied, using appropriate tools. An important consideration for grid impact studies, as the one presented in this work, is the inherently difficult validation of the obtained results. In such a framework, several aspects could contribute to deviations from reality, more so since future conditions with high heat pump and PV penetration are assumed. First, the representativeness of grid cases investigated by the framework could not be confidently claimed. This would require that they are statistically determined based on a sufficiently large sample of real networks, for which no data were available. A similar observation could be made for the buildings populating those networks. GISdata could be useful to that respect, even though usually information with regard

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to construction properties is missing. Second, the verification of load profiles is difficult, since few data are available for comparison at the spatial and temporal resolution of the impact analysis. More specifically, the load profiles of individual buildings, as well as the simultaneity of those, is very important when evaluating grid impacts. As such, annual consumption data for average households have little information to offer. Average synthetic load profiles based on the country level consumption are also not sufficient to evaluate the peaks and simultaneity at the scale of a distribution island. For this work, the used model for occupancy and domestic base-load profiles has been verified with reference values in terms of annual loads, simultaneity and autocorrelation [13]. A last point regarding validation concerns the accuracy of simulation tools, in particular the building simulation model and the network models. In this thesis, both were taken from the IDEAS Modelica library, which has been validated in Refs. [110, 225]. This work focused on air-source heat pumps and PVs as the two LCTs most adopted in dwellings to reduce energy use and improve energy efficiency. Other LCTs could be included in this analysis, such as micro-combined heat and power (CHP) and ground-source heat pumps, even though these were shown to have smaller impacts [12, 157, 158]. Electric vehicles, for instance, could have important impacts on the grid [158], and they are also highly likely to increase in number in the future. Except for other technologies, different options could be investigated for heat pumps and PV. Currently the same technical characteristics were used for all instances of heat pumps and PVs. Other options for PV could include three-phase connections, allowing larger capacity systems, or technologies with higher efficiency. Regarding heat pumps, floor heating systems and thermal storage for space heating could be additionally investigated. As more technologies and options are added to the problem, more simulations are needed to cover the potential combinations. Therefore, these options should be considered carefully, possibly with assumptions defining specific combinations. In order to tackle computational restrictions and allow more options to be included in the framework, further research could be directed to reducing the number of required evaluations, as well as shortening simulation time per evaluation. Section 3.5 investigated some options towards that direction, for instance looking into network model simplifications and analyzing the impact of the number of replications and design points. With regard to simulation speed, solutions could be explored to improve numerical efficiency of the Modelica models, for instance by utilizing different solvers, as proposed by Jorissen [109, Ch. 3]. Furthermore, high performance computing could offer important benefits for probabilistic assessment [173]. In terms of reducing the required number of evaluations, different sampling schemes could be investigated. For instance, adaptive sampling could help allocate the simulation budget more efficiently by utilizing information from previous evaluations [32, 70].


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Also due to computational restrictions, building and grid simulations have been decoupled for the grid impact analysis. This is equivalent to assuming that network conditions would not affect the operation of household equipment. In fact, PV inverters need to be disconnected in case voltage at the grid connection exceeds an upper limit [212]. This effect has been included in the model with a simple rule-based control that disregards generated power for a given amount of time after disconnection. However, in case of heat pumps, shut down due to undervoltage in the grid would affect the load profile, since thermal comfort would decrease, potentially requiring more power once the heat pump starts again. Capturing this effect without simultaneous building and network simulations would be very difficult, however calculation time increases dramatically for combined simulations [12, p. 36]. Since the loads are pre-simulated, in the proposed framework, evaluation of control strategies based on network-related signals is generally not possible. A potential solution would be to first run a network simulation as normal, based on the network conditions define some constraints for each building, re-simulate buildings with those constraints, and then re-run the network simulation. However, this approach would also be too cumbersome for a probabilistic approach where thousands of network simulations are performed. Nevertheless, simple what if scenarios could be investigated, for instance assuming that all buildings independently try to minimize their peak demand, or follow a pre-specified price signal. In this case, the input profiles for the network simulations would only need to be changed for this scenario. Possible improvements to the framework could be done with respect to the definition of loads. For example, the base-load profiles used in this work consider appliances and occupant activities based on a 2005 time-use survey, and only include active power requirements [13]. More recent data could be used to update the model, potentially taking into account predictions for future appliance efficiencies and habits. Reactive power consumption information, not only for appliances, but also for heat pumps, could also improve the accuracy of calculated impacts, as demonstrated in Section 4.8. As mentioned already, real consumption data with high spatial and temporal resolution could help further verify the fidelity of load profiles used by the framework. Except for the loads of individual buildings, considerations at the neighborhood level could be made. For instance, the framework takes into account that buildings within a neighborhood may have similar construction qualities, such that they may all be newly constructed. Similar assumptions could be made with regard to household types, defining cases with a majority of elderly people, etc. Potentially higher simultaneity could result from such clustering. Furthermore, the framework currently only considers single-family dwellings. Especially in the urban context, buildings with more families, or with commercial use could be connected on the same network. It would be particularly interesting to examine combining different

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Conclusion

uses, since this could affect simultaneity and potentially increase neighborhoodlevel self-consumption. Therefore, future research could investigate how these different loads can be identified and included in the framework. The developed framework aimed to estimate impacts on LV networks of varying size and characteristics. Few grid parameters were, therefore, selected and varied according to assumptions on their possible ranges based on relevant studies [12, 75]. However, to improve the generalization capabilities of the framework, a more thorough analysis of real networks would be beneficial for determining different network typologies, with regard to their configuration and equipment. Clustering techniques could be used for that purpose, although, this would require that such information is available in a format that allows easy manipulation, for instance GIS-based. The latter format could also provide information on the type of buildings connected to each network, at the same time providing information on how the loads should be allocated. Concerning metamodeling, this thesis only explored linear models and neural networks in detail. While these methods already showed promising results, other techniques, such as those reviewed in Section 5.7, could potentially bring advantages. Regardless of the fitting technique, metamodels would become more attractive tools if complemented with a graphical user interface that allows to easily make predictions and interactively visualize results. Metamodels of grid impact indicators, as proposed by this thesis, could be of assistance in several applications where heat pumps and PV are thought to play an important role.

List of Abbreviations BAK back-up heating CHP combined heat and power CI confidence interval CV cross-validation DHW domestic hot water DSM demand side management DSO distribution system operator I/O input/output IID independently and identically distributed IPCC Intergovernmental Panel on Climate Change LCT low-carbon technology (e.g. heat pumps and PV systems) LHD Latin hypercube design LR linear regression LV low-voltage MAE maximum absolute error MARS multivariate adaptive regression splines MSE mean squared error

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List of Abbreviations

MV medium-voltage NN neural network OLS ordinary least squares PV photovoltaic, photovoltaic system RBF radial basis function RMS root mean square RMSE root mean squared error SVR support vector regression

List of Symbols Greek Symbols α

Regression coefficient

α∗i

Lagrangian multiplier in support vector regression model

αi

Lagrangian multiplier in support vector regression model

β

Coefficients of MARS model

β

Regression coefficient

γi

Value assigned to leaf (partition) R i of input space in regression tree model

γj

Ratio of grid cases with their annual indicator occuring in week j

∆t

Temporal resolution for load profiles and grid simulation output (min)

ε

Error term in linear regression model

ε

Parameter of support vector regression specifying the tolerable error margin

ζ

cv

Mean of the average test performances of all cross-validation iterations R ×K for model validation

cv ζr,k

Test performance of best model in repetition r and iteration k of crossvalidation for model validation

ζjms ,k 0

Test performance of model j in cross-validation iteration k 0 for model selection

ζjms

Average test performance of model j in cross-validation for model selection

ζtest

Test performance of final model on independent test dataset

η

Output of neuron in neural network

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List of Symbols

θj

Correlation parameter in Kriging model

κ

Knot of hinge basis function of MARS model

λ

Regularization parameter of ridge regression model

µ

Population mean

ρj

Correlation between annual values of an indicator and the values in week j , over all grid cases

ρx y|z

Partial correlation coefficient of x and y conditional on z

ρx y

Correlation coefficient between variables x and y

σ

Population standard deviation

σ

Scale parameter of Gaussian radial basis function

φ(x, x 0 ) Correlation function between two points x and x 0 in Kriging model φ(·)

Activation function in neural network output neuron

χi

Exponent of Kriging correlation function

ψ(·)

Activation function in neural network hidden neuron

ψ(·)

Basis function in radial basis function network

Roman Symbols A

Floor area (m2 )

a

Sample statistic

a i∗

Sample statistic computed for bootstrap sample i

a∗

Averaged sample statistic over all bootstrap samples

B

Number of bootstrap samples

b

Bias parameter in support vector regression model

ci

n rbf -dimensional vector representing the center of radial basis function i

cos φ

Heat pump power factor

Ca

Cable strength (moderate, strong)

C

Parameter of support vector regression determining the smoothness of the fit

List of Symbols

D

Dataset for metamodel cross-validation

dt

Averaging period for peak component loading indicators (min)

d

Degree of polynomial in linear regression model

E

Dataset for metamodel independent testing

E d,net

Feeder annual electricity net demand (MWh)

Ed

Feeder annual electricity demand (MWh)

Eg

Feeder annual electricity generation (MWh)

E t,bf

Distribution island annual electricity back-feeding (MWh)

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E t,d,net Distribution island annual electricity net demand (MWh) E t,d,net Distribution island annual electricity net demand (MWh) f (x)

Linear function of variables x, in Kriging model

F (x)

Cummulative distribution function of variable x

F −1 (x) Inverse distribution function of variable x GHI

Global horizontal irradiation (kWh/m2 per hour or per day)

g

Factors (parameters) in probabilistic simulation experimental design

h(·)

Basis function in MARS model

i

General-purpose index

i 1...d

Indexes for predictors in terms of polynomial models, taking values between one and q

I (·)

Indicator function which equals one when its argument is true, and zero otherwise

I max

Feeder cable maximum current, 30-min averaged (A or pu)

dt I max

Feeder cable maximum current, dt -min averaged (A or pu)

J

Number of candidate models with different hyperparameters

j

General-purpose index

j best

Model with best performance in cross-validation for model selection

K

Number of folds in cross-validation for model validation

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List of Symbols

k

Index for cross-validation iteration

k

Number of factors (parameters) in probabilistic simulation experimental design

K0

Number of folds in cross-validation for model selection

k0

Index for inner cross-validation iteration

ks

Load simultaneity factor (-)

l avg

Average cable length between building connections (m)

l ex

Random exceptional length between building connections (m)

L

Dataset for metamodel training in cross-validation iteration for model validation

L0

Dataset for metamodel training in cross-validation iteration for model selection

l 1...d

Indexes for power of polynomial models, taking values between zero and d , with their sum less or equal to d

m

Number of replications per design point

n

Number of design points, feeder cases, or generally sample size

N

Number of buildings in feeder

Nt

Number of buildings in the entire distribution island

NI

Number of buildings in the distribution island, except in feeder of interest

n ach

Air change rate for ventilation and infiltration (h-1 )

n bf

Number of basis functions in MARS model

n load

Number of loads in calculation of simultaneity factor k s

nn

Number of neurons in hidden layer of multilayer perceptron neural network

n rbf

Number of basis functions in radial basis function network

n spl

Number of splits in regression tree

ori

Orientation

P 0,PV

Rooftop PV system rated capacity (kW)

pˆ

Predicted probability of success of a binary outcome in Logistic regression

List of Symbols

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p

Probability of success of a binary outcome in Logistic regression

Pd

Feeder peak electricity demand, 30-min averaged (kW)

P d,net

Feeder peak electricity net demand, 30-min averaged (kW)

Pg

Feeder peak electricity generation, 30-min averaged (kW)

pi

i th percentile of distribution

Pt

Distribution island signed peak load, 30-min averaged (kW)

P t,bf

Distribution island peak electricity back-feeding, 30-min averaged (kW)

P t,d,net Distribution island peak electricity net demand, 30-min averaged (kW) P viol

Proportion of replications with minimum voltage violating the lower limit of 0.85 pu

Q

General construction quality if the neighborhood (new, renovated, old)

q

Number of features in metamodel or predictors in linear regression model

r

Index for cross-validation repetitions

R

Number of cross-validation repetitions

Ri

Partition (leaf ) of input space in regression tree model

R2

Coefficient of determination

r HP

Heat pump penetration level in feeder (%)

r HP,I

Heat pump penetration level in distribution island, except in feeder of interest (%)

r HP,t

Heat pump penetration level in the entire distribution island (%)

r PV

PV penetration level in feeder (%)

r PV,I

PV penetration level in distribution island, except in feeder of interest (%)

r PV,t

PV penetration level in the entire distribution island (%)

RMSE j Root mean squared error between annual values of an indicator and the values in week j , over all grid cases S max

Distribution transformer maximum apparent power, 30-min averaged (kVA or pu)

216 |

List of Symbols

dt S max

Distribution transformer maximum apparent power, dt -min averaged (kVA or pu)

Sn

Distribution transformer rated capacity (kVA)

std(·)

Standard deviation of variable ·

std(P cv ) Standard deviation of the average test performances of all cross-validation repetitions R for model validation std∗ (a) Bootstrap standard deviation of sample statistic a T

Neighborhood type (rural, urban)

Tsup

Nominal supply temperature for space heating ( °C)

T

Dataset for metamodel testing in cross-validation iteration for model validation

T0

Dataset for metamodel testing in cross-validation iteration for model selection

Tamb

Ambient temperature ( °C)

Uavg

Area-weighted average U -value of the entire building envelope (W/m2 K)

Umax

Maximum feeder 10-min averaged RMS phase voltage (V or pu)

Umin

Minimum feeder 10-min averaged RMS phase voltage (V or pu)

95%w Umin Minimum 5th weekly percentile of 10-min averaged RMS phase voltage (V or pu)

Un

Nominal low-voltage phase voltage (230 V)

Uop

U -value of building envelope opaque elements (W/m2 K)

Uref

Reference voltage at the transformer secondary at no-load conditions (V or pu)

v

Function input

w

Weights between connections in neural network

wwr

Window-to-wall ratio

x

Feature of metamodel, predictor in linear regression model

y

Response variable in linear regression model or other metamodel

List of Symbols

|

217

yˆ

Predicted response from linear regression model

y

Mean value of the response variable

z

Simulation output

Z (·)

Random process in Kriging model

Zviol

Binary response classifying a design point as critical for minimum voltage violation or not

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List of publications Articles in international journals P ROTOPAPADAKI , C. AND S AELENS , D. Heat pump and PV impact on residential low-voltage distribution grids as a function of building and district properties. Applied Energy, 192:268–281, 2017. doi: 10.1016/j.apenergy.2016.11.103 PATTEEUW, D., R EYNDERS , G., B RUNINX , K., P ROTOPAPADAKI , C., D ELARUE , E., D’ HAESELEER , W., S AE LENS , D., AND H ELSEN , L. CO2-abatement cost of residential heat pumps with active demand response: Demand- and supply-side effects. Applied Energy, 156:490–501, 2015. doi: 10.1016/j.apenergy.2015.07.038 R EINBOLD, V., P ROTOPAPADAKI , C., TAVELLA , J.- P., AND S AELENS , D. Assessing scalability of a low-voltage distribution grid co-simulation through Functional Mockup Interface. Journal of Building Performance Simulation, 2018. Submitted for publication P ROTOPAPADAKI , C. AND S AELENS , D. Towards metamodeling the neighborhood-level grid impact of low-carbon technologies. Building and Environment, 2018. Submitted for publication

Peer reviewed papers at international conferences P ROTOPAPADAKI , C. AND S AELENS , D. Sensitivity of low-voltage grid impact indicators to weather conditions in residential district energy modeling. In 2018 Building Performance Modeling Conference and SimBuild co-organized by ASHRAE and IBPSA-USA, pp. 1–8, Chicago, IL, September 26-28, 2018 P ROTOPAPADAKI , C. AND S AELENS , D. Sensitivity of low-voltage grid impact indicators to modeling assumptions and boundary conditions in residential district energy modeling. In BS2017, 15th Conference of International Building Performance Simulation Association, pp. 752–760, San Francisco, 7-9 August, 2017 P ROTOPAPADAKI , C. AND S AELENS , D. Metamodeling energy indicators in neighborhoods with growing deployment of heat pumps and rooftop photovoltaics. Energy Procedia, 132:555–560, 2017. doi: 10.1016/j.egypro.2017.09.736 P ROTOPAPADAKI , C., B AETENS , R., AND S AELENS , D. Exploring the impact of heat pump-based dwelling design on the low-voltage distribution grid. In BS2015, 14th Conference of International Building Performance Simulation Association, pp. 2530–2537, Hyderabad, December 7-9, 2015 P ROTOPAPADAKI , C., R EYNDERS , G., AND S AELENS , D. Bottom-up modelling of the Belgian residential building stock: impact of building stock descriptions. In SSB2014, 9th International Conference on System Simulation in Buildings, Liege, Belgium, December 10-12, 2014

FACULTY OF ENGINEERING SCIENCE DEPARTMENT OF CIVIL ENGINEERING BUILDING PHYSICS SECTION Kasteelpark Arenberg 40 - box 2447 B-3001 Leuven [email protected] https://bwk.kuleuven.be/bwf

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