Document [original]

A Contribution to the Simplified Determination of Heat

Distribution Costs in Linear and Radial District Heating

Networks

vorgelegt von

Dipl.-Ing.

Max Bachmann

an der Fakultät III - Prozesswissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

– Dr.-Ing. –

genehmigte Dissertation

Promotionsausschuss:

Prof. Dr. Stefan Elbel

Prof. Dr.-Ing. Martin Kriegel

Prof. Dr.-Ing. habil. Joachim Seifert

Vorsitzender:

Gutachter:

Gutachter: Prof. Dr.-Ing. Christoph Nytsch-Geusen

Tag der wissenschaftlichen Aussprache: 29. November 2023

Berlin 2024

“The only way to make sense out of change is to plunge into it, move with it, and

join the dance.” – Allan Watts

Danksagung

An dieser Stelle möchte ich all jenen danken, die mich auf dem Weg zum erfolgreichen

Abschluss dieser Arbeit begleitet haben. Die vorliegende Dissertationsschrift wurde haupt-

sächlich während meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Hermann-Rietschel-

Institut (HRI) der TU Berlin erstellt. Daher gilt mein besonderer Dank zuallererst Prof.

Dr.-Ing. Martin Kriegel, dem Leiter des HRI, der es mir ermöglichte, an zahlreichen span-

nenden und lehrreichen Projekten zu arbeiten. Er hat mich sowohl fachlich als auch per-

sönlich weiterentwickelt und ich schätze sein entgegengebrachtes Vertrauen und die kon-

struktiven Besprechungen, die letztendlich zur Findung des Themas und zum erfolgreichen

Abschluss dieser Arbeit geführt haben.

Des Weiteren danke ich Prof. Dr.-Ing. habil. Joachim Seifert herzlich für sein Interesse

an dieser Arbeit und die Übernahme des zweiten Gutachtens. Ich bin ihm besonders dank-

bar für die zahlreichen Konsultationen, fachlichen Diskussionen und wertvollen Hinweise

zum Manuskript.

Für die Übernahme des dritten Gutachtens gilt mein besonderer Dank Prof. Dr.-Ing.

Nytsch-Geusen.

Außerdem möchte ich all meinen Kolleginnen und Kollegen am HRI danken, die mir

sowohl fachlich als auch persönlich zur Seite standen. Aufgrund der Vielzahl ist es un-

möglich, alle namentlich zu erwähnen. An dieser Stelle möchte ich jedoch Yannick, Stefan,

Dennis, Tunc, Lukas, Diana, Maximilian, Alexander, Raik, Anne, Michael, Eugen, Nils,

Kevin, Lucas, Karsten, Tong, Julia, Gerrid und Ferdinand besonders hervorheben. Die

zahlreichen Kaffeeunterhaltungen, fachlichen Diskussionen und Kickerrunden werde ich in

wunderbarer Erinnerung behalten.

Für die sprachliche Durchsicht und die wertvollen Hinweise zum Manuskript danke

ich besonders meinen Freunden Dr.-Ing. Dustin Ahrendt, Dr.-Ing. Eric Bach und Leo

Keilmann.

Nicht zuletzt möchte ich meinen Freunden und meiner Familie für ihre emotionale

und persönliche Unterstützung danken. Mein herzlichster Dank geht besonders an meine

Eltern Heidi und Torsten, an meine verstorbenen Großeltern Christine, Klaus, Hilde und

Rudi sowie an meine Freunde Tim, Flo, Niene, Jo, Suzy, Martin, Andreas, Kristina und

David. Insbesondere möchte ich meiner Freundin Medje ganz besonders für ihre Liebe,

Geduld, Entbehrungen und die umfangreiche Unterstützung im Alltag danken.

Berlin, im Sommer 2023

Max Bachmann

Zusammenfassung

Wärmenetze bieten eine ideale Möglichkeit einen wesentlichen Beitrag zum Gelingen der

Energiewende zu leisten, sofern eine CO2arme Wärmebereitstellung realisiert wird. Wär-

menetze haben Vorteile gegenüber der gebäudeindividuellen Wärmeversorung, wie nied-

rigere Wärmeerzeugungskosten durch Skaleneffekte, höhere Eﬀizienzen bei der Wärme-

erzeugung und die Möglichkeit zur Integration von Wärme aus verschiedenen Quellen.

Jedoch sind Wärmenetze nicht bedingungslos vorteilhaft, denn im Gegensatz zur gebäu-

deindividuellen Wärmeversorgung benötigen sie eine Investition in die Netzinfrastruktur.

Ob diese Investition die ökonomischen Vorteile auf der Seite der Wärmebereitstellung auf-

wiegt, hängt von einer Vielzahl von Faktoren ab und kann nicht prinzipiell beantwortet

werden.

In der Literatur werden zwei Methoden beschrieben, um die Eignung eines Gebiets für

die Fernwärmeversorgung zu ermitteln. Die Nutzung von detaillierten Netzsimulationen

ermöglicht eine genaue Bestimmung der Wärmeverteilungskosten, erfordert jedoch erheb-

liche Fachkenntnisse und Aufwand bei der Anwendung. Territoriale Bewertungsmethoden

basieren hingegen auf leicht ermittelbaren Indikatoren der Stadtplanung, jedoch ist ihre

Genauigkeit begrenzt. Darüber hinaus greifen territoriale Bewertungsmethoden zumeist

auf Daten bereits realisierter Wärmenetze zurück, wodurch eine Anwendung dieser Me-

thoden bei zukünftige Wärmenetzgenerationen fraglich ist.

Daher zielt diese Arbeit darauf ab, eine Methodik zur Abschätzung der Wärmever-

teilungskosten zu entwickeln, die sich nicht auf Auslegungsparameter bereits realisierter

Wärmenetze stützt. Diese Methodik soll die Vorteile der territorialen Bewertungsmethoden

(einfache Anwendung) und Vorteile der detaillierten Netzsimulation (Genauigkeit) kombi-

nieren. Dazu sollen die Haupteinflussgrößen ermittelt werden, um anschließend mehrere

Methoden abzuleiten, anhand derer sich die Kosten zukünftiger Wärmenetze vereinfacht

bestimmen lassen. Darüber hinaus soll die Genauigkeit dieser Methoden bestimmt und

deren Limitierung ermittelt werden.

Dabei konzentriert sich diese Arbeit in erster Linie auf die Analyse der Wärmevertei-

lungskosten von Liniennetzen. Zusätzlich dazu wird ein Ansatz vorgestellt, mit dem sich

die Kosten von Strahlennetzen ermitteln lassen. Insgesamt werden die Wärmeverteilkosten

in Kapital-, Wärmeverlust-, Druckverlust- sowie Betriebs- und Wartungskosten eingeteilt

und in Form von spezifischen Wärmeverteilkosten betrachtet.

Es werden zwei unterschiedliche Konfigurationen der Verbraucherverteilung betrachtet,

wobei diese die obere und untere Grenze der Kosten in einem Liniennetz darstellen. Eine

Konfigurationen nimmt an, dass sich alle Verbraucher am Ende eines Stranges befinden

wodurch das Wärmenetz an allen Stellen einen konstanten Durchmesser aufweist. Eine

weitere Konfigurationen nimmt eine konstante Verteilung der Verbraucher entlang des

linearen Wärmenetzes an. Dies führt zu einer sukzessiven Reduktion der Rohrdurchmesser

mit steigender Netzausdehnung.

Zur Beantwortung der Forschungsfragen, wurde ein detailliertes Netzmodell entwi-

ckelt. Untersuchungen daran zeigen auf, dass die Kapital- und Druckverlustkosten mit

der Netzausdehnung anwachsen, während Wärmeverluste sowie Betriebs- und Wartungs-

kosten weitgehend unabhängig von der Netzausdehnung sind.

Zur Ermittlung der Haupteinflussgrößen wurde eine “one-factor-at-a-time”- (OFAT)

und eine Monte-Carlo-Parameterstudie mittels des Detailmodells durchgeführt. Insgesamt

wurden so vierundzwanzig Parameter untersucht. Anhand der OFAT-Parameterstudie

konnten zwei wesentliche Gruppen von Eingangsparametern identifiziert werden. Eine

Gruppe beeinflusst die Wärmeverteilkosten abhängig und eine Gruppe beeinflusst die Wär-

iii

meverteilkosten unabhängig von der Netzausdehnung.

Mittels der Ergebnisse der Monte-Carlo-Studie konnten Korrelationen zwischen den

Wärmeverteilkosten und den Eingangsparametern abgeleitet werden. Die ermittelten Kor-

relationskoeﬀizienten liegen dabei in einem Bereich von |𝜌|=0.0...0.58. Die lineare Wär-

meabnahmedichte, die Netzausdehnung und der Annuitätsfaktor weisen dabei die stärks-

ten Korrelationen bezüglich der Wärmeverteilkosten auf. Zusätzlich dazu konnten sieben

weitere Einflussfaktoren mit mittleren Korrelationskoeﬀizienten ermittelt werden, was die

Bestimmung der Wärmeverteilkosten basierend auf nur wenigen Einflussfaktoren erheb-

lich erschwert. Als einziger Eingangsparameter besitzt die lineare Wärmeabnahmedichte

einen nichtlinearen Zusammenhang bezogen auf die Wärmeverteilkosten. Alle anderen Ein-

gangsparameter weisen einen linearen Zusammenhang bezogen auf die Verteilungskosten

auf.

Die Ergebnisse der Monte-Carlo-Studie wurden dazu verwendet, ein- und mehrdimen-

sionale Regressionsmodelle abzuleiten. Diese Modelle lassen sich sehr einfach anwenden,

indem die vorliegenden Designparameter des Wärmenetzes in die ermittelten Korrelations-

gleichungen eingesetzt werden. Die mittlere relative Abweichung ist jedoch vergleichsweise

hoch. Für das beste eindimensionale Modell, das ermittelt wurde, konnte eine mittlere

relative Abweichung von NRMSE = 43.9% ermittelt werden. Diese reduziert sich auf

NRMSE =25.9%für das beste mehrdimensionale Regressionsmodell. Im Gegensatz zu

bereits bestehenden territorialen Abschätzungsmethoden, können anhand des mehrdimen-

sionalen Regressionsmodells eine Vielzahl unterschiedlicher Eingangsparameter berücksich-

tigt werden, was die Kostenabschätzung für zukünftige Wärmenetzgenerationen ermöglicht.

Um eine zuverlässige Vorhersagegenauigkeit zu erreichen, die über eine grobe Abschätzung

hinaus geht, reichen die erzielten Genauigkeiten bei Anwednung der Regressionsmodelle

jedoch nicht aus.

Zur Verbesserung der Vorhersagegenauigkeit wurde daher ein analytischer Ansatz ent-

wickelt, der auf dem detaillierten Modell basiert, jedoch ohne iterative und interpolieren-

de Berechnungsschritte auskommt. Dieser analytische Ansatz führt zu Ergebnissen, die

sehr nah an den erreichbaren Ergebnissen des detaillierten Simulationsmodells liegen. Im

Schnitt konnte die relative mittlere Abweichung auf NRMSE =1.3%reduziert werden.

Die Anwendung der analytischen Methode ist etwas komplexer als die Anwendung der

Regressionsmodelle. Diese Komplexität ist jedoch nicht mit dem Berechnungsaufwand der

detaillierten Netzsimulation vergleichbar.

Die vorliegende Dissertationsschrift schließt damit eine bestehende Lücke in der Lite-

ratur und stellt mehrere leicht anzuwendende Methoden bereit, mit denen sich die Wär-

meverteilkosten zukünftiger Wärmenetzgeneration vereinfacht bestimmen lassen. So kann

ein gewisser Beitrag zur Umsetzung der Wärmewende hin zu einer CO2-armen Wärmever-

sorgung realisiert werden.

Abstract

District heating networks offer a great opportunity to make a significant contribution to

the success of the energy transition, assuming that low-carbon heat supply is achieved.

District heating networks have advantages over individual building heat supply systems,

such as lower heat generation costs due to economies of scale, higher eﬀiciencies in heat

production, and the ability to integrate heat from various sources. However, district

heating networks are not unconditionally advantageous, as they require an investment

in network infrastructure, unlike individual building heat supply systems. Whether this

investment outweighs the economic benefits on the heat supply side depends on a variety

of factors and cannot be answered in general.

In the literature, two methods are described for assessing the suitability of an area for

district heating supply. The use of detailed network simulations allows for an accurate

determination of heat distribution costs but requires significant expertise and effort in

application. On the other hand, territorial assessment approaches rely on easily identifiable

indicators from urban planning, but their accuracy is limited. Furthermore, the later

methods often rely on data from existing district heating networks, raising doubts about

their applicability to future generations of district heating networks.

Therefore, the aim of this thesis is to develop a methodology for estimating heat

distribution costs that does not rely on design parameters from existing district heating

networks. This methodology aims to combine the advantages of territorial assessment

approaches (ease of use) and detailed network simulations (accuracy). To achieve this,

the main influencing factors will be identified to derive multiple methods for determining

the costs of future heating networks. Additionally, the accuracy and limitations of these

methods is analyzed.

This thesis primarily focuses on analyzing the heat distribution costs of linear networks.

Additionally, an approach is presented for determining the costs of radial networks. Over-

all, the heat distribution costs are divided into capital, heat loss, pressure loss, as well as

operation and maintenance costs, and are examined in the form of levelized costs of heat.

Two different configurations of consumer distributions are considered, representing the

upper and lower limits of costs in a line network. One configuration assumes that all

consumers are located at the end of a network, resulting in a constant diameter of the

heat network at all locations. Another configuration assumes a constant distribution of

consumers along the linear heat network. This leads to a successive reduction in pipe

diameter with increasing network expansion.

To address the research questions, a detailed network model was developed. Investi-

gations show that capital and pressure loss costs increase with network expansion, while

heat losses, as well as operation and maintenance costs, are largely independent of the

network expansion.

To determine the main influential factors, a one-factor-at-a-time (OFAT) and a Monte

Carlo parameter study were conducted using the detailed model. In total, twenty-four

parameters were investigated. Based on the OFAT parameter study, two significant groups

of input parameters were identified. One group influences the heat distribution costs

dependent on the network expansion, while the other group influences the heat distribution

costs independent of the network expansion.

Based on the results of the Monte Carlo study, correlations between the heat distribu-

tion costs and the input parameters were derived. The identified correlation coeﬀicients

range from |𝜌|=0.0...0.58. The linear heat density, network expansion, and annuity factor

exhibit the strongest correlations with respect to the heat distribution costs. Addition-

ally, seven other influencing factors with moderate correlation coeﬀicients were identified

significantly challenging the determination of heat distribution costs based on only a few

influencing factors. As only input parameter, the linear heat density exhibits a non-linear

relationship with respect to the heat distribution costs.

The results of the Monte Carlo study were used to derive one-dimensional and multi-

dimensional regression models. These models can be easily applied by inserting the

available design parameters of the district heating network into the derived correlation

equations. However, the normalized relative error is relatively high. For the best one-

dimensional model, a normalized relative error of NRMSE =43.9%was determined. This

reduces to NRMSE =25.9%for the best multi-dimensional regression model. Unlike ex-

isting territorial estimation methods, the multi-dimensional regression model can consider

a variety of different input parameters, enabling cost estimation for future heat network

generations. However, to achieve a reliable prediction accuracy that goes beyond rough

estimation, the achieved accuracies are not suﬀicient.

To improve the prediction accuracy, an analytical approach has been developed based

on the detailed model, but it eliminates iterative and interpolating calculation steps. This

analytical approach leads to results that are very close to the achievable results of the

detailed simulation model. On average, the normalized relative error could be reduced

to NRMSE =1.3%. The application of the analytical method is slightly more complex

than the application of the regression models, but this complexity is not comparable to

the computational effort required for the detailed network simulation.

The present doctoral thesis thus fills an existing gap in the literature and provides

several easy-to-use methods for simplified determination of the heat distribution costs of

future district heating network generations. Ultimately, the results support the transition

towards sustainable and eﬀicient heat supply systems.

Contents

Page

List of Tables x

List of Figures xii

Nomenclature xvii

1 Introduction 1

1.1 Motivation .................................... 1

1.2 LiteratureReview ................................ 2

1.2.1 Ongoing Development of District Heating Systems . . . . . . . . . . 2

1.2.2 Detailed Network Analysis Approaches . . . . . . . . . . . . . . . . . 3

1.2.3 Territorial Assessment Approaches . . . . . . . . . . . . . . . . . . . 4

1.2.4 Reported Parameters Affecting the Heat Distribution Costs . . . . . 9

1.2.5 Summary ................................. 12

1.3 ResearchQuestions................................ 12

1.4 OutlineofthisThesis .............................. 13

2 Theoretical and Conceptual Framework 15

2.1 District Heating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 District Heating Consumer . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 General Description and Terminology . . . . . . . . . . . . . . . . . 16

2.2.2 ConsumerTypes............................. 17

2.2.3 Substations................................ 18

2.2.4 Reference Prices of Substations . . . . . . . . . . . . . . . . . . . . . 20

2.2.5 Simultaneity of Multiple Consumers . . . . . . . . . . . . . . . . . . 21

2.3 District Heating Heat Generators . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 General Description and Terminology . . . . . . . . . . . . . . . . . 22

2.3.2 Types of Heat Generators . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Design of Heat Generators . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 OperationModes............................. 25

2.3.5 Economic Considerations . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 District Heating Distribution Networks . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 General Description and Terminology . . . . . . . . . . . . . . . . . 27

2.4.2 NetworkTypes.............................. 28

2.4.3 NetworkTopology ............................ 29

2.4.4 PipeSystems............................... 30

2.4.5 PressureSystem ............................. 33

2.4.6 HeatLosses................................ 35

2.4.7 Nominal Temperature Difference . . . . . . . . . . . . . . . . . . . . 38

2.5 Planning Phases of District Heating Systems . . . . . . . . . . . . . . . . . 39

2.6 Economic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.1 Heat Generation Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6.2 Heat Distribution Costs . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Pipe Distribution Configurations . . . . . . . . . . . . . . . . . . . . . . . . 45

2.8 LCOH of Radial Structured DHNs . . . . . . . . . . . . . . . . . . . . . . . 46

vii

Contents

3 Detailed Simulation Model 49

3.1 ModelDesign................................... 49

3.2 Case Studies and Input Parameters . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 ModelValidation................................. 58

3.3.1 TheDataBase .............................. 58

3.3.2 Validation Against Published Model . . . . . . . . . . . . . . . . . . 58

3.3.3 Validation Against Real Case Data . . . . . . . . . . . . . . . . . . . 61

3.4 Default Model Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 General Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.2 Influence of Non-Equally Distributed and Non-Equally Sized Con-

sumers................................... 67

3.5 OFAT Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5.1 Results of the OFAT Sensitivity Study . . . . . . . . . . . . . . . . . 70

3.5.2 Summary and Conclusion of the OFAT Sensitivity Study . . . . . . 77

3.6 MonteCarloStudy................................ 81

3.6.1 Suﬀicient Number of Cases . . . . . . . . . . . . . . . . . . . . . . . 81

3.6.2 Distribution of LCOH . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.6.3 Offset Between LCOH of SCC and DCC . . . . . . . . . . . . . . . . 85

3.6.4 Correlation of Input Parameters and LCOH . . . . . . . . . . . . . . 86

3.6.5 Estimation of the Distribution Costs Using Single Input Regression

Models .................................. 87

3.6.6 Estimation of the Distribution Costs Using Multiple Input Regres-

sionModels................................ 90

3.6.7 Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Analytical Model 93

4.1 Description of the Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.1.1 Formulation of the Pipe Friction Factor . . . . . . . . . . . . . . . . 96

4.1.2 Formulation of the Pipes Insulation Thickness . . . . . . . . . . . . . 97

4.1.3 Determination of the Network LHD . . . . . . . . . . . . . . . . . . 99

4.1.4 Effective Pipe Diameters . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.1.5 Algorithm to Estimate the LCOH Using the Analytical Model . . . 108

4.2 Comparing the Characteristic Function to the Detailed Model . . . . . . . . 112

4.3 Evaluation Against Monte Carlo Data . . . . . . . . . . . . . . . . . . . . . 114

4.4 The Accuracy of the Analytical Model in Context . . . . . . . . . . . . . . . 117

4.5 Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5 Conclusions and Outlook 121

Bibliography 125

A Appendix 133

A.1 Additional Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.1.1 Estimation Error Using a Simplified Calculation of Annual Pumping

Energy...................................133

A.1.2 Assessing the Thermal Resistances of Two Buried Pipes . . . . . . . 134

A.1.3 Inner Pipe Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.1.4 Logarithmic Mean Temperature Difference . . . . . . . . . . . . . . 136

A.1.5 Heat Losses of a Single Buried Duo Pipe . . . . . . . . . . . . . . . . 137

A.1.6 Price Conversion Rates . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.1.7 Distribution of the local, the network, and the effective network LHD137

viii

Contents

A.2 Images.......................................141

A.3 Tables .......................................150

List of Tables

Table Page

1.1 Threshold values found in the literature above which a DHN can be assumed

as economical feasible. Data partly based on Reference [45]............ 6

1.2 Characteristic plot ratio of different area types according to References [44, 16]. 7

1.3 Classification of heat distribution costs based on the heat density according to

Reference [13]. .................................... 9

2.1 Typical values of peak load, base loads and full load hours of residential and

non-residential building types according to Reference [52]............. 17

2.2 Typical energetic characteristic values of an old (1970) and a new (2020) build-

ing located in Zurich according to Reference [70].................. 18

2.3 Typical design temperatures of different heat consumers according to Reference

[71]. .......................................... 18

2.4 Regression coeﬀicients referring to Equation (2.5) to estimate specific invest-

ment costs of substations shown in Figure 2.4. .................. 21

2.5 Coeﬀicients referring to Equation (2.18) to estimate specific investment costs

of several heat generator types shown in Figure 2.9. Data based on Reference

[8]............................................ 26

2.6 Overview of typical fuel prices for industrial consumers valid per 2021 without

VAT based on Reference [78]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Overview of typical specific investment costs of pumping stations converted to

the base year 2022 using conversion rates provided in Table A.3 without VAT

according to Reference [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Overview of technical and economical model parameter of the detailed model. . 49

3.2 List of default, minimum and maximum model input parameters. . . . . . . . . 57

3.3 Matching and mismatching input parameters of the detailed and the published

model.......................................... 59

3.4 Input parameters of the use-case to validate the detailed model against the

publishedmodel.................................... 59

3.5 Error of the cost prediction using the detailed model compared to results ob-

tained from the published model. . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Required input parameter needed for a validation against real-life data. . . . . 62

3.7 Relative and absolute error between the DCC and SCC LCOH distribution.

Data obtained from the detailed model using the default parameter set. . . . . 66

3.8 Mean, standard deviation, minimum, and maximum value of of several costs

components for two different network expansion. The data was derived from

Monte Carlo parameter study with 𝑛 = 10000 and assuming a distributed

consumer configuration (DCC)............................ 84

3.9 Input parameters of the single input regression model according to Equations

(3.34) and (3.35) including reached accuracies and assuming a DCC. . . . . . . 90

4.1 Overview of technical and economical model parameter of the analytical model. 95

4.2 Regression coeﬀicients to estimate the ratio of the inner and outer pipe in-

sulation diameter based on Equation (4.6) for single KMR, MMR and PMR

pipes. ......................................... 98

List of Tables

4.3 Range of input parameters for the Monte Carlo simulation to estimate the

correction coeﬀicient 𝐾𝜓. ..............................107

4.4 Evaluation parameters of the Monte Carlo simulation to estimate 𝐾𝜓. . . . . . 107

4.5 Specific input parameters of the analytical model to compare the distribution

of the characteristic function obtained from the detailed model. . . . . . . . . . 113

4.6 Error of the shape of the characteristic function of the analytical model in

comparison of the characteristic function obtained from the detailed model for

several components of the distribution costs. . . . . . . . . . . . . . . . . . . . 113

4.7 Mean and standard deviation of evaluation parameters of comparison between

detailed and analytical model of the advanced Monte Carlo data base. . . . . . 117

A.1 Nominal parameters used to estimate the electrical pumping costs for several

networkcontrolconcepts. ..............................133

A.2 Nominal parameters to estimate electrical pumping costs for several network

control concepts. The inner and outer diameters of the pipes insulation (𝑑in

and 𝑑out) were used according to Table A.8 at IC =2. ..............135

A.3 Average annual inflation rates and price conversion rates to the base of 2022

(1st of January) for Germany and the EU according to References [108, 107]. . 138

A.4 Coeﬀicients 𝑞∗

0and 𝑞∗

1describing the local LHD according to Equation (4.9) for

the rising, constant and falling scenario. . . . . . . . . . . . . . . . . . . . . . . 138

A.5 Specific investment costs for a standard substation according to minimum re-

quirements as minimum, maximum and average values [72]. Prices are con-

verted into Euro using a conversion rate of 1CHF =0.922€[76] as valid for

2020 and converted to the base year 2022 using conversion rates provided in

Table A.3........................................150

A.6 Eﬀiciencies and economical parameter of different types of heat generators

based on References [71, 8]. Cost data were converted to the base year 2022

using conversion rates of the European Union provided in Table A.3. . . . . . . 151

A.7 Pipe and trench cost data according to Reference [70]. . . . . . . . . . . . . . . 152

A.8 Pipe sizing data of KMR pipes according to Reference [70]. Geometric proper-

ties shown in Figures 2.14a (uno) and A.5 (duo). . . . . . . . . . . . . . . . . . 153

A.9 Pipe sizing data of MMR pipes according to Reference [70]. Geometric proper-

ties shown in Figures 2.14a (uno) and A.5 (duo). . . . . . . . . . . . . . . . . . 154

A.10 Pipe sizing data of PMR pipes according to Reference [70]. Geometric proper-

ties shown in Figures 2.14a (uno) and 2.14b (duo). . . . . . . . . . . . . . . . . 154

A.11 Mean, standard deviation, minimum, and maximum value of of several costs

components for two different network expansion. The data was derived from

Monte Carlo parameter study with 𝑛=10000and assuming a single consumer

configuration (SCC)..................................155

A.12 Input parameter of the single input regression model according to Equations

3.34 and 3.35 including reached accuracies assuming a SCC. . . . . . . . . . . . 155

A.13 Model coeﬀicients of the multi input regression model for a DCC. See Equation

(3.36) for the underlying model equation. . . . . . . . . . . . . . . . . . . . . . 156

A.14 Model coeﬀicients of the multi input regression model for a SCC. See Equation

(3.36) for the underlying model equation. . . . . . . . . . . . . . . . . . . . . . 157

A.15 Permutation table of pipe friction factor coeﬀicients according to the nominal

pressure loss, the pipes roughness and the nominal temperature related to

Equation (4.2).....................................158

List of Figures

Figure Page

1.1 Qualitative comparison of available methods to asses the heat distribution costs

compared to the of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Specific investment costs and annual heat losses as function of linear heat den-

sity based on Reference [30]. Costs were converted to the base year 2022 using

conversion rates provided in Table A.3. ...................... 6

1.3 Possible network connection types of four consumers according to Reference

[68]. Heat consumers are represented by numbered circles, the heat generator

is represented by the square, and the pipes are represented by black lines. . . . 10

1.4 Heat distribution costs of different network connection types for several network

expansion according to research results of Reference [68]. ............ 11

1.5 Heat distribution costs as function of the LHD. The data was derived from a

Monte Carlo study of a detailed network analysis which will be introduced in

Chapter 3........................................ 11

2.1 A simplified illustration of a DHS with the underling energy balance. . . . . . . 15

2.2 Overview of a typical heat consumer and its components in a DHS based on

Reference [70]. .................................... 16

2.3 Simplified representation of substations providing space heating but no DHW

production. The images are based on References [73, 70]. . . . . . . . . . . . . 19

2.4 Specific investment costs of a DH substation . . . . . . . . . . . . . . . . . . . . 21

2.5 Plotted simultaneity factor for different heat consumers according to Equation

(2.7) and based on Reference [77]. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Representation of a heat generator in a DHS consisting of several heat generator

units (HGU)...................................... 22

2.7 Possible annual load duration curve of a DHN based on Reference [70]. . . . . . 24

2.8 Typical operation modes of the DHN supply temperature based on Reference

[70]. .......................................... 25

2.9 Specific heat generator investment cost . . . . . . . . . . . . . . . . . . . . . . . 26

2.10 Simplified representation of a district heating network. . . . . . . . . . . . . . . 27

2.11 Different DHN types based on References [70, 81]. . . . . . . . . . . . . . . . . 28

2.12 Different network topologies of DHNs based on [80, 65]. . . . . . . . . . . . . . 29

2.13 Available nominal diameters, temperature and pressure resistance of pipe sys-

tems based on Reference [65]............................. 30

2.14 Typical structure of a heat network pipe in radial direction based on Reference

[82]. .......................................... 30

2.15 Specific pipe investment costs of a two-pipe-system normalized to the trench

length based on Reference [70] and converted to the base year 2022 using con-

version rates provided in Table A.3. ........................ 31

2.16 Specific trench costs bases on Reference [70] and converted to the base year

2022 using conversion rates provided in Table A.3. ................ 32

2.17 Maximum feasible flow velocity as function of the inner pipe diameter based

on Reference [70].................................... 32

2.18 Characteristic pressure distribution of a DHN. . . . . . . . . . . . . . . . . . . 34

2.19 Characteristic differential pressure to volume flow curve of the network and

pump for uncontrolled, constant and proportional pressure differential control. 35

xii

List of Figures

2.20 Geometric characterization for heat loss estimations of two single buried pipes

based on Reference [82]. Inner and outer diameters refer to the pipe insulation. 37

2.21 Nominal temperature difference as function of the nominal supply temperature.

Case study data was derived from Reference [30].................. 38

2.22 Planning phases to realize the heat distribution of a DHS based on References

[70,52]. ........................................ 39

2.23 Linear relationship between inner pipe diameter and specific pipe investment

costs for underground installation of KMR pipes with insulation class 2. The

data is based on prices from [68] and has been converted to the base year 2022

using conversion rates provided in Table A.3. ................... 43

2.24 Example branched DHN consisting of a single heat generator, 36 consumer,

one main branch and six sub-branches. . . . . . . . . . . . . . . . . . . . . . . . 46

2.25 Graphical explanation of the principle of separation. . . . . . . . . . . . . . . . 47

3.1 Model structure of the detailed model . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Program flow chart of the detailed simulation model. . . . . . . . . . . . . . . . 51

3.3 Distribution of the LCOH related to the network expansion of the detailed and

the published model according to the validation use-case. . . . . . . . . . . . . 60

3.4 Pipe friction factor and inner pipe diameter of published and and use case . . . 61

3.5 Specific investment costs for several values of the linear heat density (LHD)

obtained from the detailed model compared to real case data. Real case data

was derived from Reference [30]. .......................... 63

3.6 Percentage annual heat losses with respect to the LHD obtained from the de-

tailed model and compared to real case data. Real case data was derived from

Reference [30]. .................................... 64

3.7 Distribution of the LCOH related to the network expansion derived from the

default parameter set of the detailed model. . . . . . . . . . . . . . . . . . . . . 65

3.8 Distribution of the nominal consumer power related to the network expansion. 68

3.9 Variation of the nominal consumer power with respect to the network expansion

of the scenario linear changing nominal consumer power. . . . . . . . . . . . . . 68

3.10 Results of the investigations according to non-equally distributed and non-

equallysizedconsumers................................ 69

3.11 Inner pipe diameter as a function of the network length . . . . . . . . . . . . . 70

3.12 Results of the sensitivity study using the data obtained from the OFAT param-

eter study. The data was derived from evaluation the distribution costs at a

network expansion of 𝐿bra =10km assuming a DCC. . . . . . . . . . . . . . . . 71

3.13 Distribution of the total LCOH related to the network expansion. Data was

derived from the OFAT parameter study assuming a DCC part I. Black: default,

dark gray: minimum, light gray: maximum. . . . . . . . . . . . . . . . . . . . . 78

3.14 Distribution of the total LCOH related to the network expansion. Data was de-

rived from the OFAT parameter study assuming a DCC part II. Black: default,

dark gray: minimum, light gray: maximum. . . . . . . . . . . . . . . . . . . . . 79

3.15 Distribution of the total LCOH related to the network expansion. Data was

derived from the OFAT parameter study assuming a DCC part III. Black:

default, dark gray: minimum, light gray: maximum. . . . . . . . . . . . . . . . 80

3.16 Evaluation of mean value 𝜇∗and standard deviation 𝜎∗with increasing cases

of the Monte Carlo simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.18 Histograms showing LCOH components of the Monte Carlo simulation. . . . . 83

3.19 Relative offset between the total LCOH assuming a SCC and a DCC for several

network expansions 𝐿bra according to 𝑋=(𝑐dst,scc −𝑐dst,dcc)/𝑐dst,scc. ...... 85

xiii

List of Figures

3.20 Heat map of the Pearson correlation coeﬀicient for several components of the

distribution costs (total, capital, heat loss, pumping) and input parameters.

Data was derived from the Monte Carlo parameter study assuming a DCC. . . 86

3.21 Scatter plots including linear and non-linear regression of the relevant input

parameters for correlations coeﬀicients |𝜌|≥0.1. Data was derived from the

Monte Carlo parameter study assuming a DCC................... 89

3.22 RMSE and NRMSE of the multiple input regression model. . . . . . . . . . . . 91

4.1 Model structure and graphical explanation of the corresponding linear DHN of

theanalyticalmodel.................................. 94

4.2 Composition of the heat distribution costs of the analytical model. Cost com-

ponents marked by a diamond need to be provided by the analytical model. . . 95

4.3 Pipe friction factor related to the volume flow rate for several sets of input

parameters....................................... 97

4.4 Ratio of outer and inner insulation diameter related to the inner pipe diameter

valid for KMR-uno pipes. Data points were derived from Table A.8. . . . . . . 98

4.5 Graphical explanation of the connection between the consumer heat demand,

the average LHD of the network, the local LHD, and its linear regression related

to the network expansion 𝐿bra. ........................... 99

4.6 Diameter distribution of a SCC, a DCC and an effective DCC for several net-

workexpansions. ...................................101

4.7 Results of the Monte Carlo simulation to estimate the correction coeﬀicient 𝐾𝜓.108

4.8 Linear distribution of local LHD to validate the distribution of the character-

istic function of the analytical model against the results obtained from the

detailedmodel.....................................112

4.9 Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a rising distribution of the local LHD

assumingaDCC....................................114

4.10 Distribution of the nominal consumer power of the detailed model along the

linear distribution path according the regular and advanced Monte Carlo data

base...........................................115

4.11 Comparison of the results obtained from the detailed and the analytical model

according the advanced Monte Carlo data base. Number of bins per histogram

is100..........................................116

4.12 Prediction error of several cost prediction methods in comparison to the costs

obtained from the detailed model using the advanced Monte Carlo data base. . 119

A.1 Temperature and pressure control concepts used to investigate their influence

on the specific pumping costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.2 Annual power consumption for pumping of the exemplary DHN control concepts.134

A.3 Thermal resistances 𝑅1–𝑅3as function of the nominal pipe diameter and the

layingdepth. .....................................135

A.4 Temperature heating surfaces diagram. . . . . . . . . . . . . . . . . . . . . . . . 136

A.5 Geometric characterization for heat loss estimations of a single buried duo pipe

based on Reference [82]. Inner and outer diameters refer to the pipe insulation. 137

A.6 Comparison of the distribution of the local, the network and the effective LHD

for several scenarios of the local LHD 𝑞lin,loc. ...................139

A.7 Heat load and temperature profile of an exemplary heat consumer. Heat load

profile has derived using the tool TEASER [109]. The temperature profile of

a typical year was used for climate region 4 (Potsdam) according to Reference

[110]...........................................141

xiv

List of Figures

A.8 Results of the sensitivity study using the data obtained from the OFAT param-

eter study. The data was derived from evaluation the distribution costs at a

network expansion of 𝐿bra =10km assuming a SCC. . . . . . . . . . . . . . . . 142

A.9 Histograms with 100bins of LCOH components. The data was derived from

the Monte Carlo parameter study (𝑛=10000) for a comparable short (𝐿bra =

0.5km) and a fully expanded distribution network (𝐿bra =10km) assuming a

SCC...........................................143

A.10 Logarithmic histograms with 100bins of LCOH components of Monte Carlo

parameter study for a comparable short and long distribution network of a SCC.143

A.11 Logarithmic histograms with 100bins of LCOH components of Monte Carlo

parameter study for a comparable short and long distribution network assuming

aDCC. ........................................144

A.12 Heat map of the Pearson correlation coeﬀicient for several components of the

distribution costs and input parameters. Data was derived from the Monte

Carlo parameter study assuming a SCC.......................144

A.13 Scatter plots including linear and non-linear regression of the relevant input

parameters for correlations coeﬀicients |𝜌|≥0.1. Data was derived from the

Monte Carlo parameter study assuming a SCC...................145

A.14 Scatter plots and linear regression of the Monte Carlo simulation results for

cases |𝜌|<0.1and consumer heat load, consumer full load hours, interest rate

and invest horizon assuming a DCC.........................146

A.15 Scatter plots and linear regression of the Monte Carlo simulation results for

cases |𝜌|<0.1and consumer heat load, consumer full load hours, interest rate

and invest horizon assuming a SCC. ........................147

A.16 Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a constant distribution of the local LHD

assumingaDCC....................................148

A.17 Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a falling distribution of the local LHD

assumingaDCC....................................148

A.18 Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a rising distribution of the local LHD

assumingaDCC....................................149

A.19 RMSE and NRMSE of the multiple input regression model. The data was

obtained form the Monte Carlo simulation assuming a SCC............149

Nomenclature

Acronyms

Acronym Meaning/description

BG bio gas

CHP combined heat and power

COP coeﬀicient of performance

DCC distributed consumer configuration

DH district heating

DHN district heating network

DHS district heating system

DHW domestic hot water

DN nominal pipe size

EU European Union

GIS geographical information system

HGU heat generator unit

HOP heat only plant

HP heat pump

HRI Hermann-Rietschel-Institut

KMR rigid plastic jacket pipes with steel-medium pipe

LCOH levelized costs of heat

LHD linear heat density

MinOP minimum operating pressure

MIP maximum incremental pressure

MMR flexible plastic jacket pipes with steel-medium pipe

MOP maximum operating pressure

N/A not available

NG natural gas

NRMSE normalized root mean squared error

O&M operation and maintenance

OFAT one factor at a time

PMR flexible plastic medium pipe

PN nominal pressure

RMSE root mean squared error

SCC single consumer configuration

SH space heating

SPF seasonal performance factor

VAT value-added tax

VDI Verband Deutscher Ingenieure

xvii

Latin Symbols

Symbol Unit Meaning/description

𝑎1/a annuity

𝐴m2area

𝐵∗−diameter coeﬀicient

𝑐 €/kWh or €/kW specific costs

𝑐∗€/kWh specific energy price

𝑐 €/kWh or €/kW specific unit costs

𝐶 € total costs

COP coeﬀicient of performance

𝑐pkJ/kg/K specific heat capacity at constant pressure

CR −conversion rate

𝑑m diameter

𝐷 − ratio of outer and inner pipe insulation diameter

DOV −degree of variability

𝑓∗1 pipe friction factor

𝑓∗

01 pipe friction factor correlation coeﬀicient

𝑓∗

11 linear pipe friction factor correlation coeﬀicient

𝑔m/s2gravitational constant

𝑄kWh heat

ℎm height

𝑖 − counting variable

𝑖∗−counting variable

𝐼0€/m constant piping cost coeﬀicient

𝐼1€/m2linear piping cost coeﬀicient

IC −insulation class

InR %/a inflation rate

IR %/a interest rate

𝑗 − counting variable

𝐾 − coeﬀicient

𝑘 − counting variable

𝐿m length

𝑚 kg/s mass flow rate

𝑛 − amount or number

NRMSE %nominal root mean squared error

𝑃kW power

𝑝Pa pressure

PR −plot ratio



𝑄kW heat flow rate & heating capacity

𝑞kWh/m2specific heat demand

𝑞∗

0MWh/m/a constant coeﬀicient of the local linear heat density

𝑞∗

1MWh/m2/a linear coeﬀicient of the local linear heat density

𝑞 kW/m2specific heat flux

𝑟m roughness

𝑅K/W thermal resistance

𝑅2−coeﬀicient of determination

Re −Reynolds number

RES −residuum

xviii

Greek Symbols

Symbol Unit Meaning/description

RMSE ct/kWh root mean squared error

SF −simultaneity factor

SPF −seasonal performance factor

𝑇K Kelvin temperature

𝑣m/s velocity



𝑉m3/s volume flow rate

𝑊kWh work

𝑤m effective width

𝑋 − unspecified variable

𝑌 − unspecified variable

Greek Symbols

Symbol Unit Meaning/description

𝛼W/m2/K convective heat transfer coeﬀicient

Δ − difference

𝜂 − eﬀiciency

𝜆W/m/K thermal conductivity

𝜇Pa ⋅s dynamic viscosity

𝜇∗−mean

𝜈 − degree of quality

𝜙 − ratio of outer insulation pipe diameter to inner insu-

lation diameter

𝜓 − ratio of effective diameters for investment cost esti-

mation

𝜚kg/m3density

𝜌 − correlation coeﬀicient

𝜎∗−standard deviation

𝜏s time

𝜗∘C Celsius temperature

𝜉 − pressure loss coeﬀicient of fixtures

𝜁 − annual eﬀiciency

Indices

Index Meaning/description

amb ambient

am analytical model

a annual

approx approximated

A area

aux auxiliary

bra branch

bui buildings

cap capital

c Carnot

xix

Indices

Index Meaning/description

con consumer

cc consumer configuration

cor corrected

dcc distributed consumer configuration

dpt dependent

dep depth

dhn district heating network

dst distribution

DHS district heating system

eff effective

el electrical

fix fix

f fuel

full fullload

hg heat generator

hp heat pump

hyd hydraulic

i counting varible

𝑖∗counting varible

ipt independent

ins insulation

inter interaction

in internal

inv invest

j counting varible

k counting varible

L land

lin linear

loc local

loss loss

low lower

max maximum

m mean

min minimaum

𝜇mean

n nominal

nw network

op operation

om operation and maintannce

out outer

pip pipe

pri primary

prd production

𝜓index of psi

pump pumping

rad radial

real reality

ref reference

R resting

Indices

Index Meaning/description

r return

scc single consumer configuration

sec secondary

seg segments

𝜎standard deviation

soil soil

sst substation

s supply

sur surface

th thermal

tot total

trm transmission

up upper

var variable

X unspecified variable

Y unspecified variable

xxi

1 Introduction

1.1 Motivation

In 2015, the Paris Agreement was ratified by 196 countries with the objective of limiting

global warming to well below two degrees Celsius above pre-industrial levels [1]. In order

to comply with the Paris Agreement, the European Union (EU) approved the European

Green Deal in 2020, which encompasses a series of initiatives aimed at transforming the EU

into a sustainable economy, with a massive reduction in CO2emissions [2]. The building

sector alone is responsible for approximately 36%of the EU’s CO2emissions, with 80%of

those emissions being related to heating and cooling [3]. Reducing the energy consumption

of buildings or utilizing renewable energy sources for heating are two possible solutions to

decrease CO2emissions resulting from building heating. The use of renewable energy can

be achieved through building individual heating system or district heating system (DHS).

But since the heating sector is mainly based on fossil fuel today, a massive transformation

is to be expected in the upcoming years.

DHS may have major advantages compared to building individual heating solutions

when it comes to heat supply of entire districts. These include, for example, lower heat

generation costs due to economies of scale, higher eﬀiciencies in energy conversion pro-

cesses, high flexibility in the use of different heat sources, a design-related heat storage

potential, possible integration of fluctuating renewable energy sources, and the utilization

of surplus heat from power generation [4, 5, 6, 7, 8]. However, DHS are not uncondi-

tionally advantageous, when compared to building individual heating systems. This is

because DHS require investments in the necessary heat distribution infrastructure and

its maintenance, and a compensation of heat and pressure losses [9, 10, 7]. Thus, the

successful implementation of a DHS largely depends on economic considerations related

to the individual case.

In general, it can be said that the costs of DHS are typicality lower in areas with higher

heat densities [11, 12, 13]. As a result, DHS are often established in densely populated

areas where the conditions for district heating are favorable [14]. Due to a continuous

renovation process and the construction of new buildings meeting higher energy standards,

the heat demand of buildings and therefore the heat density in urban areas is expected

to decrease in the upcoming decades [6]. Moderate and progressive scenarios assume a

reduction of the heat demand between 25%[15] and 60%[5]. Nevertheless, it is assumed

that also in future scenarios, the heat densities in European urban regions will be large

enough to operate DHS eﬀiciently [5, 16]. Möller et al. [17] point out that up to 71%of

buildings heat demand can be met with district heating in urban areas of which only a

small part is used today. However, DHS can also be suitable for heat supply in sparsely

populated areas, with careful planning [11, 18]. In this context, Büchele et al. [13] figured

out that particular for small networks the network capacity leads to larger variations in

the distribution costs.

When assessing the costs of a DHS, it is useful to distinguish between heat generation

costs and heat distribution costs [18]. The cost of heat generation can be estimated with

reasonable accuracy if the type and size of the heat generator are known, with economies

of scale resulting in lower costs for larger heat generators [19, 4]. Heat distribution costs,

on the other hand, are subject to economies of scope, meaning they are heavily influenced

by site-specific conditions [20]. Two main approaches for estimating heat distribution

1. Introduction

costs can be found in the literature: detailed network analysis and territorial assessment

approach [21].

Detailed network analysis involves estimating the distribution costs using a simulation

that considers the actual network topology. This approach is expected to yield to more

accurate results since it uses more detailed information about the future district heating

network (DHN). However, it requires a high degree of in-depth knowledge and information

about the future DHN, leading to higher complexity and effort, and requiring special tool

knowledge of the user. In contrast, territorial approaches correlate heat distribution costs

with urbanistic parameters such as heat density or number of buildings, making it simpler

to use, but it may lack accuracy.

Thus, this thesis aims to develop a method that combines the benefits of detailed

network analysis and the territorial approach. Ideally, this method should be user-friendly,

similar to the territorial approaches, but offer comparable accuracy compared to a detailed

network analysis. In the context of the ongoing transformation of the energy sector, this

method is intended to be applicable to any system design parameters that may be used in

future DHSs. Figure 1.1 illustrates the objective of this thesis in comparison to detailed

network analysis and the territorial approach.

Accuracy

Complexity / eﬀort

Territorial

approach

Detailed

network

analysis

Aim of

this

thesis

Figure 1.1: Qualitative comparison of available methods to asses the heat distribution

costs compared to the of this thesis.

1.2 Literature Review

1.2.1 Ongoing Development of District Heating Systems

DHS have been used for heat supply for more than a century and are under constant

development. Today, up to five generations of DHS can be found in the literature [6, 22],

whereas the 4th and 5th generations represent development of the near future. According

to Lund et al. [6], increasing generations of DHSs are characterized by reduced system

temperatures and losses. The first generation of DHSs was mainly constructed between

1880–1930 and is characterized by very high system temperatures of up to 200°C, using

steam as heat transfer medium [6]. DHSs of the second generation were mainly used from

1930–1980 and use pressurized hot water as heat transfer medium at system temperatures

greater than 100°C [6]. Most DHSs that are in use today belong to the third generation [6,

23]. These systems are characterized by a construction year after 1980 and use pressurized

water at system temperatures mainly around 100°C [6]. The first three DHS generations

were developed to be supplied by mainly fossil fuels [24] but can also be supplied by small

shares of renewable energies [6].

1.2. Literature Review

In contrast, the 4th and 5th generations of DHSs are specifically designed to provide

decarbonized heat. It is expected that these generations will contribute the most to

the future energy system [6, 23]. According to Lund et al. [6], 4th generation DHS

can be characterized by system temperatures below 70°C, the ability to supply existing,

renovated, and new buildings with heat for space heating (SH) and domestic hot water

(DHW), low heat losses, the ability to integrate low-temperature waste and renewable

heat sources, being part of a holistic smart energy system, and ensuring suitable planning,

cost, and motivation structures. In contrast, the definition of 5th generation DHSs is

less clear and still under debate [24]. Nevertheless, the following characterizations can be

found in the literature: each consumer can operate as a producer, yielding a bi-directional

network structure [25]; utilization of the synergies of combined heating and cooling [24];

system temperatures below 30°C [24]; warm and cold pipe system temperatures are free-

floating [22]; a decentralized heat producer structure exists [24, 25]; the possibility to

use uninsulated pipes; and the usage of decentralized heat pumps in sub-stations [23, 26].

Which of the two approaches will prevail cannot be answered from today’s perspective.

However, publications show that both approaches are being implemented and are likely to

be part of the energy system in the near future [27, 25, 24, 28, 26, 29].

As the transformation of DHSs continues, it is becoming increasingly evident that

future DHS designs and operations will differ significantly from those of the 3rd generation.

One of the most notable changes will be the lower supply temperature, which is expected

to reduce the temperature difference between the supply and return pipes [30]. This

reduction in temperature spread necessitates the use of larger pipe diameters, resulting in

higher piping costs [25]. Additionally, the increased volume flow rate at lower temperature

spreads may result in higher pumping costs [25]. In cases where booster technologies are

required in substations, higher investment and electricity costs for the substations may

also be expected [25, 23]. However, lower system temperatures may also have a positive

impact on future DHS costs, as they reduce the requirements for pipe systems. This means

that less expensive pipe systems and laying techniques may be used [25]. Moreover, lower

system temperatures are expected to significantly reduce thermal losses [6, 25], and the

costs of heat generation may be reduced if energetic synergies can be exploited [27].

1.2.2 Detailed Network Analysis Approaches

As discussed in Section 1.1, two main principles, detailed network analysis and the terri-

torial approach, can be used for district heating cost assessment.

The detailed network analysis approach involves sizing the DHN according to the

actual spatial structure of the network, starting from the heat demand of individual or

clustered buildings [21]. The topological design of the network can be achieved using

mathematical optimization algorithms or expert experience, often implemented in software

tools. The distribution costs can be obtained by allocating the costs of the designed

network components, such as pipes or substations.

In terms of cost optimization for DHNs, Dorfner et al. [31] presented a method that

optimizes the topological structure and pipe diameters to minimize heat distribution costs.

Thus, their analysis indicates that pipe length, network layout, and pipe diameters are

critical parameters that affect the heat distribution costs.

Chinese [32] proposed a method to find cost-effective solutions for district heating and

cooling systems network design. The study identified heating demand profiles and energy

prices as parameters that affect the optimal solution.

Li et al. [33] introduced a genetic spatial optimization algorithm to determine the

optimal configuration for DHNs. They concluded that several factors determine the opti-

mal solution, including the heat demand of the consumers, the distance between the heat

1. Introduction

generator and the customers, and the pressure and temperature limitations.

Bordin et al. [34] introduced an optimization technique to maximize revenues and

minimize infrastructure costs while adding new users to an existing DHN. Their model

considered the law of mass continuity and a pressure loss component, and they used steady-

state conditions to simulate the heat load reduction due to the simultaneous operation of

several consumers in the network.

Falke et al. [35] proposed a heuristic approach that generates DHN configurations

randomly, from which the least costly variant is selected to obtain an economically optimal

solution. Their network optimization approach is embedded within a multi-dimensional

optimization problem, minimizing both costs and CO2emissions. It is important to note

that the proposed method was only applied to radial DHNs, and meshed networks were

not considered in their analysis.

Unternährer et al. [36] introduced a novel method for estimating heat distribution costs

in DHNs. This approach employs georeferenced data of buildings, resource availability,

and road networks to model the DHN. To reduce computational complexity, the authors

aggregate multiple buildings into spatial clusters and use the local road networks and a

graph theory-based routing algorithm to optimize the DHN layout.

Jebamalai et al. [37] proposed a GIS1-based tool that automates network routing

and pipe dimension optimization. Their tool optimizes distribution costs by considering

network routing and pipe design. To reduce computational effort, they used a network

clustering method to minimize the number of consumers. In a case study, the authors

identified network pressure levels, substation sizes, and network dimensions as influential

parameters in network costs.

A similar approach was introduced by Nielsen et al. [38], who developed a GIS-based

tool that optimizes the network layout, designs the network, and calculates the actual

distribution costs of the considered network area. They also present a case study com-

paring the district heating expansion potential with and without heat savings. The study

shows that 3rd DHS expansion is generally not economically feasible, but introducing

fourth-generation DHS significantly increases the district heating (DH) expansion poten-

tial significantly.

In addition to scientific publications on optimization methods, various software tools

are available to perform DHN optimizations. Examples of such tools include OPTIT™[39],

Comsof Heat™[40], Termis™[41] and THERMOS [42]. These tools take georeferenced

data as input and provide detailed information on the optimal network typology, required

network components and associated costs. However, they require expertise and certain

information on the future network from the user, and current computational capacities

limit the detailed analysis and optimization of larger cities [36]. Therefore, clustering

methods are often used to reduce the computational complexity [43, 36, 37].

1.2.3 Territorial Assessment Approaches

In order to perform a cost assessment of DHNs at larger scales using a more coarse ter-

ritorial approach, urbanistic parameters are commonly used to find correlations to heat

distribution costs. In the literature, several approaches can be found.

Threshold Value Approach

One simple territorial approach is the threshold method, which compares the heat demand

or heat capacity densities in a certain area to a threshold value that determines whether the

area is economically feasible for heat supply using a DHS. The threshold values are typically

1geographical information system (GIS)

1.2. Literature Review

based on previous project experiences. In the literature, three assessment parameters have

been identified: the heat density 𝑞A(MWh/m2/a), the LHD 𝑞lin (MWh/m/a), and the

linear heat load density 𝑞lin (kW/m/a).

The heat density parameter relates the annual heat demand of an area 𝑄A,ato the

land area 𝐴L(see Equation (1.1)). This parameter can be easily calculated if the heat

demand of an area is known. However, it does not consider the fact that the costs of a

DHN is usually affected by the length of the network. In contrast, the LHD parameter

corresponds to the ratio of the annual heat demand in an area to the total network trench

length (see Equation (1.2)). The LHD is commonly used in the literature as an economic

evaluation parameter for DHS [44, 12, 45, 23]. The linear load capacity density parameter,

which relates the overall nominal heat capacity in an area to the network trench length (see

Equation (1.3)), can also be used to evaluate the economics of DHS. Both the LHD and

the linear load density require information on the total trench length of an area, which can

be obtained from additional parameters such as the effective width, introduced by Werner

in 1997 [46].

𝑞A=𝑄A,a

𝐴L

(1.1)

𝑞lin =𝑄A,a

𝐿(1.2)

𝑞lin =

𝑄n

𝐿(1.3)

Table 1.1 provides an overview of threshold values for assessing the economic feasibility

of implementing DHS in a given area, as reported in the literature. Among the fourteen

references surveyed, ten suggest using the LHD values as the evaluation parameter, while

two suggest using the heat density and two suggest using the linear capacity density.

However, using the LHD values as evaluation parameter poses a challenge, particularly in

areas with lower values, as the threshold values range from 0.2...2.0MWh/m/a. Moreover,

the threshold values reported in the literature are solely based on average cost data of

realized DHS projects, considering both heat generation and heat distribution costs. This

approach should be critically evaluated, especially in the context of the ongoing develop-

ment of DHS, particularly those of the 4th and 5th generation (as discussed in Section

1.2.1). Additionally, the use of threshold values provides binary information on whether

an area is suitable for DHS implementation or not, without providing any information on

expected distribution costs. This poses diﬀiculties if either the heat generation or heat

distribution costs deviate significantly from the values used to define the threshold val-

ues. Furthermore, the threshold value approach does not consider other network design

or economic parameters, such as temperature spread or annuity.

Linear Heat Density Approach

In addition to the threshold value approach, several references provide more detailed in-

formation on the actual heat distribution costs [30, 30, 12, 53]. Specifically, Reference [30]

provides specific investment of district heating networks as a function of the LHD. Guide-

line and real case data are given in Figure 1.2a. The guideline data shows that specific

investment costs of DHNs tend to increase with decreasing values of the LHD. However,

the impact of low LHD values on investment costs may vary depending on the pipe system

used and the conditions at hand. The real case data shown in Figure 1.2a also indicate a

significant impact of LHD on specific investment costs, with a spread of data indicating

that other variables may also affect the economic feasibility of a district heating network.

1. Introduction

Table 1.1: Threshold values found in the literature above which a DHN can be assumed

as economical feasible. Data partly based on Reference [45].

Threshold value Year Reference Additional notes

𝑞lin >0.2MWh/m/a 2011 [47] -

𝑞lin >0.2...0.3MWh/m/a 2006 [48] -

𝑞lin >0.3MWh/m/a2008 [49] -

𝑞lin >0.9MWh/m/a 2009 [50] -

𝑞lin >1.0MWh/m/a 2011 [12] Seasonal operation and only SH

𝑞lin >1.2MWh/m/a 1999 [51] -

𝑞lin >1.5MWh/m/a 2016 [9] -

𝑞lin >2.0MWh/m/a 2018 [52] -

𝑞lin >2.0MWh/m/a 2011 [12] All year operation SH & DHW

𝑞lin >2.0MWh/m/a 2022 [53] -

𝑞A>10.0kWh/m2/a 2008 [49] -

𝑞A>15.0kWh/m2/a 2011 [54] -

𝑞lin >1.0kW/m/a 2018 [52] -

𝑞lin >1.2kW/m/a 2014 [55] -

Heat losses are another factor affecting the economy of district heating systems. Figure

1.2b shows the annual heat losses as a function of the LHD, with a clear trend of higher

relative heat losses expected at low values of the LHD.

012345

Linear heat density [MWh/m/a]

200

400

600

800

1000

1200

1400

Specific invest costs of heat

distribution [€/(MWh/a)]

Guidline KMR bad

Guideline KMR good

Guideline PMR

Real cases (incl. SST)

Real cases (excl. SST)

Maximum acceptable

(a) Network investment costs.

0 1 2 3 4 5

Linear heat density [MWh/m/a]

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

Annual heat losses [%]

Guideline

Real cases

(b) Heat losses of DHNs with 𝜗s,n= 90°C,

𝜗r,n=70°C and annual operation.

Figure 1.2: Specific investment costs and annual heat losses as function of linear heat den-

sity based on Reference [30]. Costs were converted to the base year 2022 using conversion

rates provided in Table A.3.

Effective Width Approach

Persson et al. [44] proposed a method for assessing heat distribution costs of DHNs at

larger scales using urbanistic parameters. This method was revised in 2019 [16] and 2021

[56]. The authors focused solely on capital costs and neglected additional costs for pressure

and heat losses as well as for maintenance. To estimate the heat distribution costs, the

authors described the specific heat distribution capital costs 𝑐cap using Equation (1.4),

where 𝐶inv represents the network investment costs, 𝑎represents the annuity factor, and

𝑄con,arepresents the annual heat sold to consumers. The network investment costs are

1.2. Literature Review

described using a linear relation with 𝐼0as the constant, 𝐼1as the slope, and 𝑑in as the

average internal pipe diameter of the network.

𝑐cap =𝑎𝐶inv

𝑄con,a=𝑎𝐿(𝐼0+𝐼1𝑑in)

𝑄con,a=𝑎(𝐼0+𝐼1𝑑in)

𝑞lin

(1.4)

To relate the LHD to urban parameters, the effective width (𝑤) concept was introduced

by Persson et al. This theoretical measure relates the land area 𝐴to the network length2

𝐿, as shown in Equation (1.5). The authors analyzed 73 Swedish DHNs and established

a connection between the effective width and the plot ratio PR, as expressed in Equation

(1.6). The plot ratio is a commonly used parameter in urban planning and is defined as

the ratio of building floor area to the land area. Table 1.2 presents typical values for this

parameter. Nielsen et al. [18] and Dénarié et al. [21] adapted Persson et al.’s method

using data from Denmark and Italy, respectively. Both authors found slightly different

correlations, which are presented in Equations (1.7) and (1.8). Therefore, by knowing the

annual heat demand of a land area and the effective width, the LHD of a land area can

be determined using Equation (1.9). The average internal pipe diameter is the missing

component in Equation (1.4). This value can be estimated using the LHD, as shown in

Equation (1.10). This equation was derived from 134 Swedish DHSs and city districts [44].

𝑤=𝐴

𝐿(1.5)

𝑤Sweden =61.8PR−0.15 (1.6)

𝑤Denmark =80.144𝑒−0.744PR (1.7)

𝑤Italy =50.25PR−0.127 (1.8)

𝑞lin =𝑄con,a𝑤

𝐴(1.9)

𝑑in

m=0.0486ln (𝑞lin

GJ/m)+0.0007 (1.10)

The effective width approach has been widely employed in recent literature to evaluate

the potential suitability of areas for district heating supply [47, 18, 57, 58, 59, 21, 56, 60,

61]. This method has been applied at various scales, ranging from a city [57, 21], to a

country [18, 59, 61, 60], and even a European level [58, 16]. The effective width approach

provides a rough indication of which areas may be suitable for DH development and which

may not. The method is attractive due to its simplicity, transparency, replicability, and

the fact that it requires only heat density and plot ratio maps, both of which can be

obtained from open source data platforms [62]. However, the accuracy of the method

is a concern. Fallahnejad et al. [62] compared distribution costs estimated using the

effective width approach with those derived from detailed network analysis in 15 city sub-

regions and found that the effective width approach overestimated the heat distribution

costs by approximately 44%in their example. However, the degree of underestimation

or overestimation may vary depending on the specific example considered. Moreover,

2trench length of the network

Table 1.2: Characteristic plot ratio of different area types according to References [44, 16].

Area characteristic Inner city Outer city Park areas

Plot ratio 0.5≤PR ≤2.0 0.3≤PR <0.5 0≤PR <0.3

1. Introduction

the effective width approach only considers trench length within an area and neglects the

network structure in that area. Additionally, each country requires a unique representation

of the effective width correlation [18, 21]. Furthermore, the effective width and average

pipe diameter correlation are based on empirical data obtained from completed projects,

making it challenging to estimate the heat distribution costs of 4th or 5th generation district

heating systems where network design parameters are expected to vary significantly. In

2022, Jiang et al. [23] noted that using only the LHD is no longer suﬀicient to evaluate

DH potentials. In addition to that, heat and pressure loss costs as well as operation and

maintenance (O&M) costs are not considered at all.

Further Distribution Cost Approaches

Girardin et al. [63] propose a method that shares similarities with the effective width

approach suggested by Persson et al. [44]. However, in contrast to the effective width

approach, Girardin et al. utilize an empirical approach (represented by Equation (1.11))

to estimate the length of a DHN. This method takes into account the number of buildings

𝑛bui present within a given area 𝐴and a calibration coeﬀicient 𝐾, which was derived from

data collected from existing DHNs.

𝐿=2(𝑛bui −1)𝐾√𝐴

𝑛bui

(1.11)

Dochev et al. [64] propose a novel approach to evaluate areas suitable for DH supply

based on the LHD. Unlike the effective width approach, the authors utilize the street

layout and a graph theory-based algorithm to provide information on hypothetical DHN

layouts. The heat demand information was obtained using a heat demand atlas of the city

of Hamburg. To determine the suitability of an area for DH supply, the authors applied a

threshold value of 𝑞lin ≥1.5MWh/m/a, as suggested by Dötsch et al. [65] and Jalil-Vega et

al. [66]. Therefore, this method combines the threshold value approach with information

on the actual street layout. The authors evaluated the predicted DHN lengths against

actual lengths, which showed an offset within ±3%for the main and medium-sized DHN

sections. However, for small-sized DHN sections around single-family houses, a deviation of

−13.3%was observed. In their study, only the LHD is considered as a decisive parameter

for assessing the suitability of an area for DH supply. Other DHN design parameters, such

as the network structure and the average pipe diameter, are neglected, which warrants a

critical assessment.

Büchele et al. [13] suggest a method to estimate the overall heat distribution costs

by summing the heat transmission and heat distribution costs. The transmission costs

consider network segments from a heating plant to a heat distribution area where little

to no heat extraction is realized. The authors consider investment costs for pipes and

operation and maintenance costs. The specific heat transmission costs 𝑐trm were found

to be 𝑐trm = 83.5€/km/a, with no further network design parameters considered. In

contrast to this, the authors suggest estimating the heat distribution costs 𝑐dst according

to the heat density. They differentiate the heat distribution costs according to five groups

of heat density. Their selected classification is presented in Table 1.3. As the suggested

method only considers the specific heat demand of an area without any information on

the structure of the DHN, the accuracy of this method is expected to be lower than the

accuracy of the effective width approach. It should be noted that the proposed method

does not take into account network design parameters, such as the network structure or

the average pipe diameter, which could lead to inaccurate results.

1.2. Literature Review

Table 1.3: Classification of heat distribution costs based on the heat density according to

Reference [13].

Heat density [GWh/km2/a] 0 – 10 10 – 20 20 – 35 35 – 60 > 60

𝑐dst[ct/kWh] 4.92 3.51 3.00 2.66 2.41

Sarma et al. [67] present a data-driven black box model for estimating heat distribu-

tion costs in DHNs. The authors incorporate both capital and operational costs and use

a relatively small database of 20 existing DHNs in Latvia. Three models are considered,

including a linear model with two independent variables (Lin-2), a linear model with five

independent variables (Lin-5), and a non-linear model with two independent variables

(Non-Lin-2). The independent variables considered in all three models are the total net-

work length and the maximum inner pipe diameter. Additionally, the linear model with

five independent variables considers the minimum and average pipe diameters as well as

the heat production costs. The reported accuracy of the models ranges from 3.03%for

the Lin-2 model to 14.43%for the Non-Lin-2 model. The Lin-5 model achieves an accu-

racy of 7.36%. However, the very limited dataset of only 20 DHNs introduces significant

uncertainty, making it diﬀicult to assess the general applicability of the proposed method

in the context of DHNs.

1.2.4 Reported Parameters Affecting the Heat Distribution Costs

According to the reported distribution cost assessment approaches presented in Sections

1.2.2 and 1.2.3, many influencing parameters affecting the heat distribution costs can al-

ready be stated. The detailed network approaches indicate that the network topology, the

pipeline length and the size of the network pipes have a significant influence on the heat

distribution costs. According to that, also the temperature spread between supply and

return pipe, the heat demand and nominal connection load of the consumers, the simul-

taneity of the heat load of several consumers, the investment costs of the piping system,

the labor costs for trench excavating, the investment costs of the consumer substations,

pressure and heat loss costs, the nominal design pressure loss, and further parameters

show a significant influence on the heat distribution costs.

Zinko et al. [49] analyzed the economical feasibility of DHSs in areas with low heat

densities. They come to the conclusion, that DHS can be economical feasible in areas with

heat densities 𝑞A>10kWh/m2/a or 𝑞lin >0.3MWh/m/a when the planning process is re-

alized with more careful planning. They further found out that direct connection between

the DHN and consumers, small local networks, measures that reduce piping diameters, a

high degree of connectivity of consumers, using of low cost piping systems, and no over

sizing of DHSs may reduce the heat distribution costs significantly.

Nussbaumer et al. [68] have conducted a thorough investigation on the parameters that

affect the heat distribution costs of a simple linear DHN. Both technical and economical

parameters were analyzed, including the investment costs of the piping network, labor

costs for trench excavation, the interest rate, payback time, heat generation costs, and

electricity price. Additionally, technical parameters such as the temperature difference

between supply and return, specific nominal pressure loss or the maximum allowed flow

velocity, insulation thickness and heat transfer coeﬀicient of the pipes insulation, average

temperature difference between the network and the ambient, consumer full load hours,

linear heat density, total network length, and the general structure of the distribution

network were considered.

One key finding from their analysis was that the network structure has a significant

1. Introduction

impact on heat distribution costs, even when holding the network length and consumer

capacity constant. The authors differentiated between radial and linear connections, with

linear connections being further divided into a SCC and DCC (see Figure 1.3). In all three

connection configurations, the total network length and the consumer capacity are assumed

similar but the resulting network structure differs significantly. In a radial connection, each

of the four pipe segments3connects one consumer to the heat generator. On the other

hand, in the linear SCC connection, all consumers are connected to the very end of a

linear network, resulting in a continuously sized network where each pipe segment must

be sized to accommodate the full capacity of all consumers. In contrast, in the linear

DCC connection, each consumer extracts a certain volume flow rate to meet their heat

load, resulting in a continuous reduction in pipe segment diameters downstream of the

linear network.

Linear connection DCC:

1 2 34

Linear connection SCC:

1 2 34

Radial connection:

Figure 1.3: Possible network connection types of four consumers according to Reference

[68]. Heat consumers are represented by numbered circles, the heat generator is repre-

sented by the square, and the pipes are represented by black lines.

Nussbaumer et al. [68] estimated the costs of heat distribution by assuming a uniform

number of full load hours 𝜏full =2000h/a, a uniform consumer capacity 

𝑄con =500kW,

and a uniform pipe segment length Δ𝐿=500m. They further assumed a nominal tem-

perature difference of Δ𝜗n=30K to calculate the heat distribution costs4. The resulting

costs are illustrated in Figure 1.4.

If a radial connection is utilized, the heat distribution costs are independent of the

network size. This configuration represents the ideal case, and results in the smallest aver-

age pipe diameter since each pipe segment is designed to transport the nominal capacity

of just one consumer. In contrast, for a linear SCC connection, the pipe diameters of all

segments are affected by all consumers since they are located at the very end of the linear

network. Hence, adding an additional consumer leads to increased diameters of all pipe

segments, resulting in a higher average diameter and ultimately higher heat distribution

costs. Similarly, for a linear DCC connection, adding an extra consumer increases the heat

distribution costs. However, this effect is less pronounced compared to the linear SCC.

The reason behind this is that the diameters of pipe segments further away from the heat

generator are reduced, resulting in smaller average pipe diameters and therefore smaller

heat distribution costs.

Zach et al. [7] conducted an analysis of parameters that affect the environmental and

economic feasibility of DHSs from an integrated spatial and energy planning perspective.

They utilized the fields of spatial planning, resource management, environmental planning,

and energy and building technology to perform their analysis. In total, 31 parameters were

evaluated through comprehensive literature research and discussion rounds with experts

from relevant research fields. The objective of their assessment was to identify key pa-

rameters that influence the environmental and economic feasibility of DHSs. The authors

3A pipe segment in this context consist of a parallel supply and return pipe.

4Please refer to Reference [68] for additional input parameters.

1.2. Literature Review

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Cumulated network length [m]

Heat distribution costs [ct/kWh]

Radial connection

Linear connection (SCC)

Linear connection (DCC)

Figure 1.4: Heat distribution costs of different network connection types for several net-

work expansion according to research results of Reference [68].

categorized these parameters into two groups: indicators that directly affect the environ-

mental and economic feasibility of DHSs and parameters that are system drivers. Based

on their analysis, the following four indicators are crucial system indicators: heat density

or linear heat density, consumer full load hours, temperature level, and available heat

sources. These indicators can be influenced by eight system drivers: building types in a

district, potential for compacting and extension, mix of different consumer profiles in a

district, thermal insulation potential of buildings, and density of workplaces, buildings,

and people in a district.

Some studies suggest that the heat density or resulting LHD are crucial factors affecting

the heat distribution costs [11, 44, 47, 18, 13, 16]. However, other studies point out that

this only applies to areas with higher heat densities [49, 11, 23]. If lower heat densities

are expected, such as in less densely populated areas or due to increased building energy

standards reducing the heat demand, other parameters may become more relevant [49, 11,

23]. This effect is illustrated in Figure 1.5, which shows heat distribution costs as a function

of the LHD for a wide range of DHN design parameters. The data was obtained from a

Monte Carlo study using a detailed network analysis approach that will be presented and

explained in Chapter 3. The results indicate that the variability of heat distribution costs

is low for higher values of the LHD and high for lower values, which may suggest that

other parameters may become more relevant in these cases.

0 1 2 3 4 5

Linear heat density [MWh/m/a]

Heat distribution costs

[ct/kWh]

Data

Trend (r²=0.6448)

Figure 1.5: Heat distribution costs as function of the LHD. The data was derived from a

Monte Carlo study of a detailed network analysis which will be introduced in Chapter 3.

1. Introduction

1.2.5 Summary

DHS have been in use for over a century and have undergone several generations of de-

velopment. The literature reports up to five generations of DHS. Currently, the 4th and

5th generations are being constructed and will continue to be used in the future. Future

DHS must address challenges such as reduced consumer energy demands, extrapolation

of areas with lower heat densities, integration of renewable energy sources, and a more

decentralized heat distribution structure. These developments are expected to affect the

general economical feasibility of DHS.

To asses heat distribution costs two distinct groups of methods were identified from a

literature research: the detailed network analysis and the territorial approach.

The detailed network approach is considered to be the more reliable method as it

takes into account the actual, or a realistic representation of the network structure, the

resulting piping design under technical design conditions, and the actual operation of the

DHS. This approach is not limited to data derived from previous projects, which makes

it highly capable of analyzing innovative DHS of the 4th and 5th generations. To perform

a detailed network analysis, commercial or non-commercial software tools or individually

created spreadsheets can be used. However, the downsides of this approach are that it is

time-consuming, requires significant computational effort, and is not capable of analyzing

large areas. Additionally, the person using this approach must have expert knowledge and

training skills, which increases the barrier to using the method.

In contrast to the detailed network approach, the territorial assessment approaches

reported in the literature require significantly fewer computational resources as the dis-

tribution network is mainly represented through empirical data. This method is highly

applicable to identifying larger areas where successful integration of a DHS is likely to be

economically feasible. However, the accuracy of this approach is lower compared to the

detailed network analysis method, with reported errors of up to 44%. Furthermore, the

empirical data used in territorial assessment approaches is typically derived from already

realized DHS projects that belong mainly to the 3rd generation. This makes it challenging

to apply territorial assessment approaches to 4th and 5th generation DHS without further

adaptions.

Based on the available literature, a broad range of parameters influencing the cost of

heat distribution in DHS have been identified. These include but are not limited to network

typology, pipeline length, and pipe dimensions, all of which appear to have a significant

impact on the overall heat distribution costs. The revised studies show that assessing

the heat distribution costs of a DHS is a very complex topic since the heat distribution

costs are affected by many different influencing parameters. This becomes increasingly

important when DHN are to be constructed in areas with lower heat densities.

1.3 Research Questions

Related to the literature research presented in Section 1.2, numerous factors may influence

the heat distribution costs, with heat density or LHD being the dominant factor at higher

values of the LHD. However, at lower values of the LHD, other factors seem to become

increasingly relevant regarding heat distribution costs. In addition, to date, no method is

available that combines the advantages of the detailed network approach, which provides

precision, with the advantages of the territorial assessment approach, which has low com-

putational effort and is simple to use, to assess heat distribution costs. Furthermore, the

available territorial assessment methods only consider heat density and network length,

but neglect network structure and other influencing network design parameters. This may

lead to unreasonably low accuracy, particularly for DHSs in areas with low heat demand

1.4. Outline of this Thesis

densities. With the development of future DHSs, the prediction accuracy of the available

territorial assessment methods is expected to decrease even further, as heat densities are

expected to decrease due to increasing building energy standards and exploration of pos-

sible DHS areas with low heat densities. Furthermore, the system design of future DHS

of the 4th and 5th generations is expected to be significantly different from that of the

DHSs currently in use, leading to an additional reduction in the prediction accuracy of

the current available territorial assessment approaches.

According to the gaps and considerations outlined in the literature, this thesis aims to

address the following research questions:

1. What are the primary parameters that influence the heat distribution costs of a

DHN?

2. To what extent do these parameters impact the heat distribution costs?

3. How can heat distribution costs determined precisely with reduced effort compared to

detailed network analysis techniques and increased simplicity compared to territorial

assessment methods?

4. Is it possible to utilize this method to future DHSs of the 4th and 5th generation?

5. What is the accuracy of this method?

6. What are the limitations of this method?

1.4 Outline of this Thesis

This thesis is structured into five main chapters in order to answer the research ques-

tions. Chapter 1 provides an introduction to the motivation, state of science, and research

questions. In Chapter 2, the theoretical background necessary to comprehend the basic

technical and economic aspects of DHS is presented. Additionally, a novel method is in-

troduced that facilitates the segmentation of arbitrarily complex radial DHN into linear

segments, enabling the subsequent chapters to focus exclusively on the determination of

heat distribution costs of linear DHN. In Chapter 3, a comprehensive simulation model

is presented that allows the estimation of heat distribution costs according to a detailed

network analysis method. This chapter includes the introduction of the selected model

design, validation of model accuracy against a published model and real application data,

identification of main influencing parameters that affect heat distribution costs, and their

evaluation according to a one-factor-at-a-time parameter study. A subsequent Monte

Carlo parameter study is conducted to investigate the interaction of several influencing

parameters on heat distribution costs. The results of this study are then used to derive

several regression models, which enable the prediction of heat distribution costs based on

network design parameters. Chapter 4 introduces an analytical approach to estimate heat

distribution costs, which is compared to the results obtained from the detailed simulation

model and the regression models introduced in Chapter 3. The final Chapter 5 provides

a conclusion that contextualizes the research findings to the initially formulated research

questions. Additionally, the limitations of the developed methods and recommendations

for further work are also being discussed in this chapter.

2 Theoretical and Conceptual

Framework

This chapter provides a comprehensive theoretical overview of DHSs and their correspond-

ing system equations. The typical components of a DHS are introduced and their technical

and economic properties are discussed in detail. Subsequently, the principle of separation

is presented, which facilitates the partitioning of a radial DHN into linear segments.

2.1 District Heating Systems

The most basic configuration of a DHS can be observed in Figure 2.1. Typically, a DHS

comprises three subsystems: a heat generator, a district heating network, and a number

of substations. The heat generator acts as a source of heat and raises the temperature

of the heat transfer fluid by adding the producer heat flow rate 

𝑄prd to the system. The

heat transfer fluid is then transported to the consumers through the DHN, which typically

consists of a supply and a return pipe, as well as at least one electrically-driven pump.

During the transportation process, heat and pressure losses occur. The electrical energy

from the pump compensates the pressure losses, while an increased heat flow rate from

the heat generator compensates the heat losses. The steady-state energy balance of the

entire system can be expressed by Equation (2.1).

In the upcoming sections, all three subsystems, as well as the overall DHS, are examined

in greater detail, and their underlying system equations are presented. The discussions

are confined to areas that are essential for understanding the investigations in this thesis.

For a more thorough explanation of DHS, readers are encouraged to refer to works such

as References [45, 69, 70].



𝑄prd +𝑃el =

𝑄con +

𝑄loss,s+

𝑄loss,r(2.1)

District heating system

Heat generator District heating network Substation

(consumer)

Supply pipe

Return pipe

Figure 2.1: A simplified illustration of a DHS with the underling energy balance.

2. Theoretical and Conceptual Framework

2.2 District Heating Consumer

2.2.1 General Description and Terminology

The primary function of a DHS is to supply heat to a number of distributed consumers

within a particular distribution area. As such, the consumers form the basis for the design

of any DHS. Figure 2.2 provides a graphical representation of a typical heat consumer.

DHN Connection line Cellar

line

Sub-

station

Building

distribution

Supply

Return

Primary system Secondary system

Figure 2.2: Overview of a typical heat consumer and its components in a DHS based on

Reference [70].

The connection between the DHN and the building refers to the connection line, which

is usually owned by the operator of the DHS. Inside the building, the supply and return

of the connection line are connected to the substation via the cellar line, which is usually

owned by the owner of the building. The connection between the cellar line and the

building’s heat distribution system is commonly called the energy transfer station or the

substation whereas the later term will be used in this thesis. It can be separated into a

transmission station and a heating center. Its task is to adapt the physical properties, such

as pressure, temperature, and volume flow, of the DHN to the required properties inside

the building and to measure the heat consumption [70]. The heat transmission station is

usually owned by the operator of the DHS, and the heating center is owned by the owner

of the building. Substations can be characterized by a direct or indirect connection, which

will be explained in more detail in Section 2.2.3. The heat distribution inside the building

is realized by the heat distribution system. The heating center and the heat distribution

system both refer to the secondary system. On the other hand, the connection line, cellar

line, and transmission station refer to the primary system.

The heat demand of a consumer can vary over time and can be mathematically repre-

sented by the variable 

𝑄con(𝜏). The heat load or consumer power1expected under design

conditions2is referred to as the nominal consumer power 

𝑄con,n, which is a crucial pa-

rameter for sizing the substation. The annual heat demand 𝑄con,aof a consumer can be

obtained by integrating the consumer’s power over a year (refer to Equation (2.2)).

𝑄con,a=∫1𝑎

0

𝑄con(𝜏)𝑑𝜏 (2.2)

1Heat load and consumer power express the same physically entity. In this thesis mainly the wording

consumer power is used.

2Design conditions generally define the conditions at which the highest consumer power is expected.

Nominal design temperatures are used to determine the nominal heating load of a building, which describe

the lowest ambient temperature that can be expected at the building’s location.

2.2. District Heating Consumer

The ratio of annual heat demand and the nominal consumer power leads to the full

load hours of the consumer. This value describes a rather theoretical quantity than an

actual physical state (see Equation (2.3)) and indicates how many hours a consumer would

have to run in full load to achieve its annual heat demand.

𝜏full =𝑄a



𝑄n

(2.3)

To cover the heating power of a consumer, the district heating operator needs to provide

a suﬀicient primary supply temperature and a suﬀicient pressure difference between the

supply and return pipe. The required supply temperature of a consumer depends highly

on the consumer type. See Section 2.2.2 for a detailed discussion. Each substation is

designed regarding a minimum pressure difference Δ𝑝sst,nand a certain design temperature

difference Δ𝜗sst,nbetween supply and return at design conditions.

In summary, a heat consumer can be characterized by the nominal consumer power,

the annual full load hours, the annual heat demand, the minimum pressure difference, and

the nominal design temperature difference of the substation.

2.2.2 Consumer Types

Heat consumers of district heating systems can be classified into different categories, in-

cluding residential, commercial, institutional buildings, and industrial consumers [69]. All

of these building types require either comfort heat or process heat. Comfort heat is pri-

marily utilized for space heating and DHW production [52]. Moreover, the system could

potentially be adapted to offer cooling services during the summer months, using technolo-

gies such as sorption-driven chillers. The space heating load is highly dependent on the

ambient temperature, as the heating load increases with decreasing ambient temperatures.

On the other hand, the DHW load exhibits daily and weekly variations. In contrast to

comfort heat, industrial processes have highly variable heat demands, which depend on

the specific process [70]. Table 2.1 provides an overview of the typical full load hours, as

well as specific peak and base loads of typical residential and non-residential buildings.

Table 2.1: Typical values of peak load, base loads and full load hours of residential and

non-residential building types according to Reference [52].

Building type Specific peak load Specific base load Full load hours

[kW/(MWh/a)] [kW/(MWh/a)] [h/a]

Oﬀice 0.44 0.02 2270

Restaurant 0.32 0.04 3120

Hospital 0.37 0.03 2700

School 0.44 0.02 2270

Single family house 0.45 0.01 2220

Apartment house 0.36 0.02 2800

To compare the thermal characteristics of different buildings, it is common to normalize

both the annual heat demand and nominal consumer power by a reference area, such as

the net lease area. This leads to the specific annual heat demand 𝑞con,aand the specific

nominal consumer power 𝑞con. In Table 2.2, an example of typical values for specific

heat demand, specific nominal consumer power, and full load hours of an old and a new

building is presented. As can be seen, advanced energy standards in newer buildings result

in lower energetic values for space heating. However, for DHW, the energetic values remain

constant, as they are more influenced by user behavior than by the building’s energetic

2. Theoretical and Conceptual Framework

condition. When considering the combined energetic values, it is apparent that newer

buildings have significantly lower specific consumer powers, specific annual heat demands,

and full load hours than older buildings.

Table 2.2: Typical energetic characteristic values of an old (1970) and a new (2020) build-

ing located in Zurich according to Reference [70].

Entity Unit 1970 2020

Space heating 𝑞con [W/m2] 80 20

𝑞con [kWh/m2/a] 185 20

𝜏full [h/a] 2300 1000

Domestic hot water 𝑞con [W/m2] 5 5

𝑞con [kWh/m2/a] 20 20

𝜏full [h/a] 4000 4000

Combined 𝑞con [W/m2] 85 25

𝑞con [kWh/m2/a] 205 40

𝜏full [h/a] 2400 1600

The required supply temperature of the consumer’s secondary system is determined

by the type of the consumer and its secondary design temperatures. The temperature

requirements for space heating mainly depend on the design temperatures of the building’s

heating system and the ambient temperature. Modern buildings can often be operated

with lower supply temperatures, while older buildings require higher supply temperatures

to cover their heating power. DHW consumers require a minimum supply temperature

of approximately 60°C throughout the year to prevent sanitary issues due to legionella

growth [45]. An overview of typical design temperatures of the secondary system is given

in Table 2.3.

Table 2.3: Typical design temperatures of different heat consumers according to Reference

[71].

Consumer type SH radiators

(old building)

Radiators Panel

heating

DHW Process

heat

Required nominal

temperature level

up to 90°C55...70°C35°C > 65°C30...300°C

2.2.3 Substations

The interface between the building’s heat distribution system and the DHN is facilitated

by a substation, which can be established through either a direct or indirect connection.

Figure 2.3 shows both types, but it is important to note that the examples do not encom-

pass the preparation of DHW, as they aim for simplicity. According to Reference [72], the

main components of a substation are:

• Shut-off valves to shut down the substation from the DHN in case of an emergency,

• a visual display of temperature and pressure in supply and return of the primary

system (not shown in Figure 2.3),

• a venting in the supply (top) and drainage in the return (bottom) (not shown in

Figure 2.3),

2.2. District Heating Consumer

• a strainer in the supply,

• a heat exchanger (only for indirect substations),

• a mixing circuit (only for direct substations),

• a control valve including actor,

• a heat meter including sensors,

• a pressure differential control valve,

• a controller for secondary supply temperature including sensors, and

• a pressure maintenance system (only for indirect substations).

Shutoﬀ

valves

Heat

meter

Strainer

Pressuer

diﬀerential

controller

Control

valve

Mixing

circuit

Heat

consumer

(a) Direct

Shutoﬀ

valves

Heat

meter

Strainer

Pressuer

diﬀerential

controller

Control

valve

Heat

exchanger

Pressure

maintenance

Heat

consumer

(b) Indirect

Figure 2.3: Simplified representation of substations providing space heating but no DHW

production. The images are based on References [73, 70].

In the case of a direct substation, the heat transfer fluid of the DHN is identical to

the heat transfer fluid circulating through the heat distribution system of the building.

Since no system deviation is realized between the primary and secondary side, no pres-

sure maintenance system on the secondary side is required, as the operator of the DHS

ensures the required differential pressure [70]. Additionally, no temperature reduction

occurs between the primary and secondary systems since there is no heat exchanger. This

characteristic can lead to a reduction in supply temperature of the DHN, resulting in im-

proved energy eﬀiciency. However, a disadvantage of direct substations is the possibility

of failures within the building, which could cause problems or even the breakdown of the

entire DHS. Furthermore, the secondary system requires protection against high pressures

in the DHN [69]. The stability of the heat distribution system within the building should

comply with the requirements established by the operator, thus limiting the freedom of

selecting appropriate pressure stabilization mechanisms. Load control is usually achieved

2. Theoretical and Conceptual Framework

by means of a mixing circuit and adjustment of the primary mass flow rate 𝑚pri in response

to the supply temperature of the secondary side 𝜗s,sec. The secondary supply temperature

is typically depending on the ambient temperature.

In the context of an indirect consumer station, the primary and secondary sides are

separated by a heat exchanger which results in a temperature reduction between the supply

side of the primary and secondary system. Additionally, a pressure maintenance system

is required on the secondary side making an indirect substation typically more expensive

than a direct substation [69]. However, the advantages of an indirect substation include

increased security as a leakage in the building distribution system may not result in a

leakage of the entire DHN, and a clear distinction between the building’s heat distribution

system and the DHN [69]. The pressure stability of the heat distribution system can be

selected independently of the DHN. The load control is typically performed by the primary

mass flow rate 𝑚pri based on the supply temperature of the secondary side 𝜗s,sec. The set

value of the secondary supply temperature usually depends on the ambient temperature.

For both substation types, the steady-state energy balance of the primary side of

a substation can be expressed as Equation (2.4). The term 1represents the energy

dissipation due to pressure losses, which is usually significantly smaller than the remaining

terms and can be neglected.



𝑄con +

𝑄loss +𝑚pri

𝜚(𝑝s,pri −𝑝r,pri)

⏟⏟⏟⏟⏟⏟⏟⏟⏟

= 𝑚pri (ℎs,pri −ℎr,pri)(2.4)

To ensure an appropriate district heating supply of the consumer, the following two

conditions must be met by the DHN at all times and at all locations of the network:

1. The pressure difference between supply and return on the primary side is greater

than the minimum allowed pressure difference of the substation Δ𝑝sst,min.

2. The supply temperature on the primary side is greater than the required supply tem-

perature on the secondary side plus the temperature difference in the heat exchanger

if an indirect substation is used.

The value of the minimum allowed pressure difference of the substation is defined by

the operator of the DHS. Typical values range from Δ𝑝sst,min =0.4...1.0bar [74, 75, 70].

2.2.4 Reference Prices of Substations

Similar to other areas within the energy sector, the economy of scale concept applies

to substations. This implies that the specific investment costs of substations decrease

with an increase in the connection loads [72]. Figure 2.4 presents the specific investment

costs of a substation as a function of the nominal substation connection power. The data

used for this plot is based on average prices of a typical substation provided in Reference

[72] and has been converted to € with the base year of 2022. It includes the cost of all

major components of a substation described in Section 2.2.3, excluding components of

the heat distribution system. The costs also encompass delivery, assembly, testing, and

commissioning, but do not incorporate piping costs or costs for electrical installations.

To establish a functional relationship between the nominal connection power and the

specific investment costs of a substation, a regression function of the form presented in

Equation (2.5) was derived from each set of specific investment costs. The coeﬀicients 𝐾1

and 𝐾2are given in Table 2.4.

2.2. District Heating Consumer

0 25 50 75 100 125 150 175 200

Substation nominal connection load ˙

Qsst,n[kW]

200

400

600

Specific invesment costs

ˆcinv,sst [€/kW]

Minimum

regression

R2= 0.9863

Average

regression

R2= 0.9919

Maximum

regression

R2= 0.9966

Figure 2.4: Specific investment costs of a DH substation as function of the nominal con-

nection load based on Reference [72]. Prices are converted into € using a conversion rate

of 1CHF =0.922€[76] as valid for 2020 and converted to the base year 2022 using con-

version rates provided in Table A.3.

𝑐inv,sst =𝐾1

ln(

𝑄sst,n)+𝐾2(2.5)

Table 2.4: Regression coeﬀicients referring to Equation (2.5) to estimate specific investment

costs of substations shown in Figure 2.4.

Coeﬀicient Unit Minimum Average Maximum

𝐾1[€] 183.212 247.521 362.968

𝐾2[kW] -1.813 -1.773 -1.737

2.2.5 Simultaneity of Multiple Consumers

A DHS is designed to supply heat to multiple consumers, not just one. As a result, the

maximum power of all consumers is typically lower than the sum of their nominal loads

due to the simultaneity of multiple consumers. This effect becomes more pronounced as

the number of heat consumers supplied increases. The simultaneity factor (𝑆𝐹) is used to

describe the ratio between the maximum simultaneous heat demand of all heat consumers

and the summed nominal consumer power, as given in Equation (2.6) [77]. Here, 

𝑄𝑖(𝜏max)

represents the power of consumer 𝑖at the time of the heat load peak of the entire DHN

𝜏max, while 

𝑄n,𝑖 is the nominal load of consumer 𝑖.

SF =∑𝑛con

𝑖=1 

𝑄𝑖(𝜏max)

∑𝑛con

𝑖=1 

𝑄n,𝑖 (2.6)

The simultaneity factor characterizes the effect of multiple consumers on a DHS. This

factor is typically less than 1 and decreases as the number of consumers increases. When

the consumer structure is homogeneous, an approximation suggested by Winter et al. can

be used to estimate the simultaneity factor for systems with less than 200 consumers (see

Equation (2.7) and Figure 2.5) [77]. However, for systems with more than 200 consumers,

the simultaneity factor is fixed at SF =0.47.

SF(𝑛con)=𝐾1+𝐾2

1+(𝑛con

𝐾3)𝐾4(2.7)

2. Theoretical and Conceptual Framework

Where 𝐾1=0.450,𝐾2=0.5512,𝐾3=53.844and 𝐾4=1.763.

0 25 50 75 100 125 150 175 200

Number of consumers ncon [-]

0.00

0.25

0.50

0.75

1.00

Simultanety factor SF [-]

Figure 2.5: Plotted simultaneity factor for different heat consumers according to Equation

(2.7) and based on Reference [77].

2.3 District Heating Heat Generators

2.3.1 General Description and Terminology

ADH heat generator is a crucial component of a DHS, responsible for ensuring the system’s

heat supply. Figure 2.6 presents a simplified depiction of a typical heat generator along

with its associated physical quantities. A heat generator may comprise a single unit or

multiple units connected to form an aggregate [70]. The selection of a single unit or an

aggregate depends on the specific scenario at hand.

HGU1HGU2HGU3

Heat generator Supply pipe

Return pipe

Figure 2.6: Representation of a heat generator in a DHS consisting of several heat generator

units (HGU).

The heat generator plays a crucial role in meeting the thermal load requirements of

the DHN that arise from the consumers heat demand and heat losses. To attain the

desired supply temperature 𝜗s,dhn for the DHN, the thermal load 

𝑄hg (see Equation (2.8))

is introduced into the network. Both the mass flow rate 𝑚hg of the heat generator and the

return temperature of the DHN 𝜗dhn depend on the operation mode of the DHN, the heat

losses of the DHN, and the thermal load requirements of the consumers [70]. Equation

(2.8) establishes a connection between the thermal load of the heat generator 

𝑄hg and the

input and output enthalpy flows (ℎs,dhn and ℎr,dhn) of the DHN. Depending on the type

of heat generator, different physical descriptions arise between the thermal power 

𝑄pri,

incoming electrical power 𝑃el,aux, and outgoing electrical power 𝑃el,out.



𝑄hg = 𝑚hg (ℎs,dhn −ℎr,dhn)(2.8)

2.3. District Heating Heat Generators

2.3.2 Types of Heat Generators

The classification of DHS heat generators can be based on the type of energy source and

the corresponding conversion processes. This categorization comprises of heat only plant

(HOP), combined heat and power (CHP) plants, and heat pump (HP) plants.

The operation of HOPs involves the conversion of a primary heat flow into a thermal

power 

𝑄hg, while no electrical power 𝑃el,out is produced. The process is characterized by

losses, and its performance can be quantified using Equations (2.9) and (2.10), where 𝜂th

and 𝜁th denote the current and annual thermal eﬀiciency, respectively. Both eﬀiciency

values can be related to the upper or lower heating value, which needs to be taken into

account when formulating the energy equilibrium formulation.

Additionally, HOPs typically have an auxiliary electrical demand 𝑃el,aux, which is

utilized for supporting systems. Depending on the type of system, this energy consumption

may or may not be considered in the energy balance. Gas, oil, waste, or wood boilers are

some examples of typical HOPs used for heating purposes.



𝑄pri =

𝑄hg

𝜂th

(2.9)

𝑄pri,a=𝑄hg,a

𝜁th

(2.10)

In contrast to HOPs, CHP plants convert a primary heat flow into both thermal and

electrical power, which can be mathematically expressed by Equations (2.11) and (2.12).

However, this process also exhibits losses, which can be quantified by the thermal and

electrical eﬀiciency parameters, denoted by 𝜂th and 𝜂el, or by the annual equivalents 𝜁th

and 𝜁el, if applicable. The combination of both eﬀiciencies results in the total eﬀiciency

parameter, denoted by 𝜂tot or 𝜁tot. In addition, CHP plants require a certain amount of

electricity, denoted by 𝑃el,aux, for auxiliary systems. This demand can be fulfilled directly

by the self-produced electricity. The power generation of CHP plants can be categorized as

either gross or net, while the total eﬀiciency can reach values up to 90%[71]. The electrical

eﬀiciency is influenced by the nominal electrical load and the employed technology. Steam,

gas, and combined cycle power plants are some common examples of CHP plants.



𝑄pri =

𝑄hg

𝜂th +𝑃el,out

𝜂el =

𝑄hg +𝑃el,out

𝜂tot

(2.11)

𝑄pri,a=𝑄hg,a

𝜁th +𝑊el,out,a

𝜁el =𝑄hg,a+𝑊el,out,a

𝜁tot

(2.12)

HP plants utilize a significant amount of auxiliary electrical, thermal, or mechanical

energy to increase the temperature level of low-temperature primary heat 

𝑄pri. In this

thesis, only electrically driven heat pumps are discussed, since they are commonly used.

Heat pump plants do not generate electrical power, which is why they could also be

characterized as heat-only plants. However, since the energy conversion of both types

differs significantly, they are discussed separately.

The primary heat flow 

𝑄pri and the electrical auxiliary power 𝑃el,aux are converted

into the usable thermal power 

𝑄hg of the heat generator (see Equation (2.13)). If energies

instead of energy flows are considered, Equation (2.13) can be expressed as Equation (2.14).

The eﬀiciency of this process is described by the coeﬀicient of performance (COP), which

is defined as the ratio of the usable thermal power 

𝑄hg to the electrical auxiliary power

consumption 𝑃el,aux (see Equation (2.15)). The annual eﬀiciency of an heat pump is called

the seasonal performance factor (SPF), which can be expressed by Equation (2.16).

2. Theoretical and Conceptual Framework

The COP of an heat pump can be further described by the Carnot eﬀiciency 𝜂cand

the degree of quality 𝜈hp of the heat pump, where the Carnot eﬀiciency depends on the

upper and lower temperature levels 𝑇up and 𝑇low. Equation (2.17) shows that the smaller

the difference between the upper and lower temperature levels are, the larger the COP

becomes. For this reason, heat pump plants can be advantageous in particular in low-

temperature DHS.



𝑄hg =

𝑄pri +𝑃el,aux (2.13)

𝑄hg,a=𝑄pri,a+𝑊el,aux,a(2.14)

COP =

𝑄hg

𝑃el,aux

(2.15)

SPF =𝑄hg,a

𝑊el,aux,a

(2.16)

COP =𝜂c𝜈hp =𝑇up −𝑇low

𝑇up 𝜈hp (2.17)

In Table A.6 an overview of typical eﬀiciencies of common types of heat generators is

given.

2.3.3 Design of Heat Generators

The demand for the thermal power in a DHN is not constant and varies throughout the

year. Figure 2.7a illustrates a typical example of an annual load curve3for a DHS. The

figure shows the areas of space heating, DHW, and heat losses. The demand for DHW and

heat losses remains relatively constant throughout the year. In contrast, the demand for

space heating varies significant due to fluctuations in ambient temperatures. The shape

of the annual load curve shows that the nominal power is only required for a few hours

per year. For the rest of the year, a much lower power is required.

Space heating

Domestic hot water DHN heat losses

1000

800

600

400

200

Thermal power [kW]

0100 150 200 250 300 35050

Days of a year [d/a]

(a) Separation into SH, DHW and heat losses.

Base load

Peak load

Part load of base load

1000

800

600

400

200

Thermal power [kW]

0100 150 200 250 300 35050

Days of a year [d/a]

(b) Seperation into base and peak load.

Figure 2.7: Possible annual load duration curve of a DHN based on Reference [70].

Figure 2.7b presents the annual load curve of a DHS, separated into peak and base

load. The base load is characterized by a high number of annual full load hours, whereas

the peak load exhibits a significantly lower number of annual full load hours. If the base

load heat generator is unable to fulfill the part load, the peak load heat generator can be

utilized as an alternative [70]. Consequently, a heat generator of a DHS may consist of

several units that are interconnected to form an aggregate system. An aggregated system

3The annual load curve represents the cumulative frequency of the heat generator load over the year

in terms of the number of days.

2.3. District Heating Heat Generators

enables the utilization of benefits from various types of heat generators. For instance, heat

generators with high capital and low operating costs are suitable for the baseload because

higher capital costs can be compensated by high annual full load hours when operating

costs are low. In contrast, heat generators with low capital costs are appropriate as peak-

load generators. As they operate for a short period each year, higher operating costs can

be accepted [69].

2.3.4 Operation Modes

The thermal power of a heat generator is commonly regulated to maintain a specified

supply temperature 𝜗s,dhn for the DHN. Three typical operating modes, namely constant,

gliding, and gliding-constant, can be distinguished [70] and are shown in Figure 2.8. If

the gliding mode is chosen, the supply temperature changes in relation to the ambient

temperature. At lower ambient temperatures, the supply temperature rises until the heat

capacity of the heat generator is reached, while it drops if the ambient temperature in-

creases until the heating limit is achieved. The gliding mode is generally suitable for

space heating consumers, as their heat demand scales according to the ambient tempera-

ture. The gliding operation mode results in lower fluctuations of the mass flow rate in the

DHN, as the supply temperature adjustment already provides a coarse adjustment of the

network’s heat capacity.

−20 −15 −10 −5 0 5 10 15 20

Ambient temperature [°C]

100

120

Supply temperature [°C]

Gliding

Constant

Gliding-constant

Figure 2.8: Typical operation modes of the DHN supply temperature based on Reference

[70].

The constant mode is characterized by a supply temperature that remains independent

of the ambient temperature. This allows suﬀicient heat capacity to be provided for all

types of consumers throughout the year, and the fluctuating mass flows regulate the DHN

capacity. However, due to the higher supply temperatures relative to the gliding mode,

higher heat losses occur.

The gliding-constant mode is a combination of the two other modes, in which the supply

temperature glides within a particular range of the ambient temperature, while it remains

constant outside that range. This mode is widely used and permits the simultaneous

supply of space heating, DHW, and process heat consumers.

2.3.5 Economic Considerations

The economics of a heat generators can be generally characterized as an economy of

scale, whereby larger heat generators typically exhibit lower specific investment costs [19,

8]. This effect is a consequence of the higher eﬀiciencies in energy conversion processes

2. Theoretical and Conceptual Framework

and lower specific investment costs associated with larger installations [8]. A graphical

representation of the specific investment costs of various heat generator types can be found

in Figure 2.9, where specific investment costs are plotted against installed capacity. The

figure illustrates that the principle of economy of scale is applicable to all analyzed heat

generators. The specific cost functions for each heat generator type can be expressed as a

function of the heat generator’s nominal output capacity, as indicated by Equation (2.18)

[8]. Here, 𝑋represents the nominal output power of the heat generator, which can be

expressed in terms of nominal electrical capacity 𝑃hg,nor nominal thermal capacity 

𝑄hg,n.

The corresponding coeﬀicients for each heat generator type in Figure 2.9 are listed in Table

2.5.

𝑐inv,hg =𝐾1(𝐾2

𝑋)(1−𝐾3)(2.18)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

Heat generator nominal output capacity ˙

Qhg,nor Pel,n[MW]

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Specific heat generator

investment costs ˆcinv,hg [M€/MW]

Wood chip HOP

Natural gas HOP

Electric HOP

Natural gas CHP (Turbine)

Natural gas CHP (Engine)

Bio gas CHP (Engine)

Air source HP

Exess heat HP

Figure 2.9: Specific heat generator investment cost as function of nominal heat generator

capacity. Data based on [8] and converted to the base year 2022 using conversion rates

provided in Table A.3.

Table 2.5: Coeﬀicients referring to Equation (2.18) to estimate specific investment costs

of several heat generator types shown in Figure 2.9. Data based on Reference [8].

Type 𝐾1[€/MW] 𝐾2[MW] 𝐾3[−]

Wood chip HOP 0.75 6.84 0.828

NG HOP 0.26 10.0 0.375

Electric HOP 0.26 25.0 0.703

NG CHP (turbine) 1.24 0.1 0.884

NG CHP (engine) 1.14 1.0 0.911

BG CHP (engine) 1.24 1.0 0.826

Air source HP 1.55 1.0 0.778

Excess heat source HP 1.35 1.0 0.733

Fuel costs represent a significant portion of variable costs and are based on the fuel

consumption of the plant and the prevailing fuel prices. As such, fuel costs scale with the

energy output per year but not with the installed capacity of the heat generator. Table

2.6 provides an overview of fuel prices applicable to the German market. It is important

to note that fuel prices are subject to market fluctuations and individual negotiations

between energy suppliers and consumers, and therefore up-to-date values should be used

in calculations.

2.4. District Heating Distribution Networks

The O&M costs of a heat generator can be divided into fixed and variable components.

Fixed O&M costs are typically paid per year and are dependent on the installed capacity

of the plant. These costs include all expenses that are independent of the operation

hours, such as administration, insurances, network or system charges, and property tax

[8]. Variable O&M costs, on the other hand, are dependent on the operation hours of the

plant and include the consumption of auxiliary energy and materials, disposal treatment,

spare parts, and output-related repairs and maintenance. An overview of typical economic

and technical parameters of various heat generator types, including O&M costs, can be

found in Table A.6.

Table 2.6: Overview of typical fuel prices for industrial consumers valid per 2021 without

VAT based on Reference [78].

Fuel type Natural gas Bio gas Electricity Wood chips

Fuel price 𝑐∗f[€/MWh] 46 79 167 26

In summary, the specific costs of heat generation can be effectively described using

information available in the literature. The principle of economy of scale applies, with

larger installations typically exhibiting lower specific costs.

2.4 District Heating Distribution Networks

2.4.1 General Description and Terminology

DHNs can be described as closed systems consisting of at least two insulated pipes that

carry a heat transfer fluid to distribute heat from a heat generator to consumers, as shown

in Figure 2.10. Similar to other utility services, the pipes are usually buried underground

[45]. The necessary circulation in the system is provided by a pump. Additionally, a

pressure maintenance system is employed to maintain the resting pressure and compensate

for volume changes of the heat transfer fluid. The thermal power transported through the

DHN can be calculated using Equation (2.19). Therefore, an increase in either the mass

flow rate 𝑚dhn or the temperature difference Δ𝜗dhn results in an increase in the capacity

of the DHN, and vice versa.

District heating network

Heat

generator

Pressure

maintenance

system

Return pipe

Supply pipe

Pump

Figure 2.10: Simplified representation of a district heating network located between a heat

generator and heat consumers.



𝑄dhn = 𝑚dhn (ℎs,dhn −ℎr,dhn)= 𝑚dhn 𝑐p(𝜗s,dhn −𝜗r,dhn)= 𝑚dhn 𝑐pΔ𝜗dhn (2.19)

The investment costs associated with a DHN generally constitute over 50%of the total

costs of a DHS [70]. Moreover, the economic viability of a DHN depends on the individual

2. Theoretical and Conceptual Framework

circumstances (economy of scope), rather than scaling effects (economy of size) [20]. As

such, particular attention should be given to the economic analysis of the heating network.

2.4.2 Network Types

Contemporary DHNs are commonly designed as two-pipe systems, consisting of one supply

and one return pipe (see Figure 2.11a). However, this configuration has limitations since

it can only provide one temperature level to all consumers at a given time, except in

some use-cases where certain consumers can utilize the return pipe [79]. To address this

limitation, three, four, and even multi-pipe systems have been developed and some of them

are utilized in practice.

Supply

Return

(a) Two-pipe-system

Supply II

Return

Supply I

(b) Three-pipe-system

Supply II

Supply I

Return II

Return I

Level I

HG1

HG2

Level II

Level III

(d) Multi-level-system

Figure 2.11: Different DHN types based on References [70, 81].

Three-pipe systems comprise of two supply and one return pipe (see Figure 2.11b).

The physical properties of the two supply pipes can be controlled independently, allowing

for constant temperature supply for DHW and process heat consumers as well as glid-

ing supply temperatures for space heating consumers. Consequently, the average system

temperature can be lowered, leading to reduced heat losses. However, the requirement of

an additional pipe compared to a two-pipe system leads to increased costs. Additionally,

discontinuous removal from the second supply pipe can cause mass flow fluctuations of the

return pipe, which may shift the operation point of the plant and result in an inconsistent

mass flow rate to the customer plants [65, 80, 70].

Four-pipe systems are composed of two supply and two return pipes, which can be

viewed as two parallel two-pipe systems (see Figure 2.11c). The temperature of the two

supply pipes can be independently controlled, enabling the adjustment of the supply tem-

perature to match consumer temperature demands, further reducing heat losses. However,

these systems are less common due to the high costs associated with their implementation.

Multi-pipe systems have also been reported in the literature [81]. These systems fea-

ture at least three different pipes (see Figure 2.11d), with consumers located between two

adjacent pipes with respect to pipe temperature. This configuration permits the optimiza-

tion of available heat sources and is especially useful in areas where consumers require

2.4. District Heating Distribution Networks

(a) Linear (b) Meshed

Figure 2.12: Different network topologies of DHNs based on [80, 65].

heat at multiple temperature levels and where heat supply at different temperature levels

is available.

2.4.3 Network Topology

The topology of a DHN is typically determined by urbanistic factors, such as the arrange-

ment of buildings and roads, the distribution area’s size, and the number of heat generators

[65]. Two types of network topologies can be distinguished, which are radial and meshed

networks [80].

Radial DHNs consist of multiple network branches that branch off from the heat gen-

erator without creating a loop (see Figure 2.12c). Therefore, only one possible path exists

from the heat generator to the consumers. This topology is vulnerable to disruptions

on that path, as any interruption in one branch would disconnect all consumers located

behind it from the heat generator’s supply [65]. However, radial DHNs have the shortest

network lengths and decreasing pipe diameters starting from the heat generator [65]. Con-

sequently, this network topology has lower investment costs compared to meshed DHNs.

Radial DHNs are generally recommended for small and medium-sized networks consist-

ing of a single heat generator [65]. A special type of a radial DHN is the linear network,

which has a single branch and relatively short connection lines to the consumer (see Figure

2.12a).

On the other hand, meshed DHNs are characterized by several loops (see Figure 2.12b).

Therefore, in most sections of the network, there is more than a single possible path from

the heat generator to the consumers. As a result, this network topology is less vulnerable

to network interruption. Meshed DHNs are commonly used in large supply areas with

multiple heat generators. However, due to their longer network lengths, meshed networks

2. Theoretical and Conceptual Framework

are typically more expensive than branched networks [65]. Purely meshed networks are

rarely found, as branched areas are often found in the outer periphery [80]. A special type

of a meshed DHN is a ring network, which consists only of a single loop (see Figure 2.12d)

[80].

2.4.4 Pipe Systems

In DH applications, three types of pipes are commonly used, which are rigid plastic jacket

pipes with steel-medium pipe (KMR), flexible plastic medium pipe (PMR), and flexible

plastic jacket pipes with steel-medium pipe (MMR). These pipes are distinguished based

on their intended use, as illustrated in Figure 2.13. Both single and twin pipe (Duo)

variants are available for each type, but the latter is typically limited to smaller nominal

diameters [65]. Compared to single pipes, twin systems exhibit marginally reduced heat

losses and lower costs [68]. KMR pipes are supplied as bars, whereas MMR and PMR

pipes are delivered in coils for smaller pipe diameters [65].

Temperature and

pressure resistance

Nominal diameter

DN 50 DN 100 DN 150 DN 1000

KMR

Single

MMR

Duo & single Single

Duo & single

PMR

Duo & single Single

140 °C

25 bar

120/130 °C

16/25 bar

95 °C

6 bar

Figure 2.13: Available nominal diameters, temperature and pressure resistance of pipe

systems based on Reference [65].

The typical structure off all three pipes types is identical. The heat transfer fluid

is carried by a carrier pipe, which is cased by a thermal insulation layer often made of

polyurethane foam and an outer jacket usually made of plastic.

The carrier pipe in KMR pipes is typically composed of steel, resulting in a pipe system

that possesses high temperature and pressure resistance, thereby making it suitable for

Jacket pipe

Insulation

Carrier pipe

(a) Single

Jacket pipe

Insulation

Carrier pipe

(b) Duo

Figure 2.14: Typical structure of a heat network pipe in radial direction based on Reference

[82].

2.4. District Heating Distribution Networks

use in various types of DHNs. KMR pipes are supplied in bars and must be assembled

on-site (i.e. welded), thereby increasing installation costs [65]. Nevertheless, this pipe

system is widely used in DH applications due to its standardization, low material costs,

and robustness [65].

In contrast, MMR and PMR are flexible and therefore self-compensating pipe systems.

MMR is composed of a flexible metallic carrier pipe, while PMR consists of a plastic

carrier pipe. The availability of these pipe systems on reels facilitates installation and can

reduce installation costs [68]. However, their pressure and temperature resistance is lower

compared to KMR pipes, which explains why flexible pipe systems are typically used for

branch and house connection pipes [68].

Typically, DHN installations employ different pipe systems for various sections, such as

main, branch, and house connections [68]. Besides the type of pipe system used, heat net-

works can also be distinguished based on pipe installation. Overground and underground

installations are possible, but overground installations are not commonly used outside of

industrial areas due to aesthetic considerations [70]. Underground installations can be

performed in trenches and shafts. However, shaft construction is expensive, making un-

derground installations in trenches more common [70]. Thus, the focus in this thesis will

be on underground installations in trenches.

The cost of the piping system can be broken down into two main components: the

capital costs of the pipes themselves and the labor costs required to install the pipes

underground within trenches. In Reference [70], an overview of the specific pipe costs per

meter of trench length for various typical pipe systems is provided, as shown in Figure

2.15. These costs include all expenses associated with materials and installation, such as

pipes, bends, tees, sockets, strain zones, pipe supports, weld material, monitoring system,

and pressure tests for a two-pipe-system. However, the costs do not account for the X-ray

of welds, relocation of utility lines, or any necessary traﬀic regulations [70]. It should be

noted that the figures provided serve as guidelines and that the specific costs may vary

depending on the individual case.

20 25 32 40 50 65 80 100 125 150 200 250

Nominal pipe diameter DN

500

1000

1500

Specific pipe costs ˆcpipe [€/m]

KMR - Single

MMR - Single

PMR - Single

KMR - Duo

MMR - Duo

PMR - Duo

Figure 2.15: Specific pipe investment costs of a two-pipe-system normalized to the trench

length based on Reference [70] and converted to the base year 2022 using conversion rates

provided in Table A.3.

As reported by Reference [70], the expenses associated with excavation works comprise

the excavation process itself, the removal of 30%of the excavated material, the placement

of a sand bed, the installation of piping, the backfilling of the trench, and the restoration

of the surface. It should be noted that trenching costs may vary depending on the location

of the installation, as stated in [70]. Figure 2.16 provides an overview of trenching costs

in relation to the nominal pipe diameter for installations located in an open field and in a

street. The figure indicates that specific trenching costs generally increase as the nominal

2. Theoretical and Conceptual Framework

pipe diameter rises above DN 50. Moreover, trenching works conducted in an open field

are about half as costly as those executed within a street. Additionally, the costs presented

in the graph are estimations only and may differ from case to case. The overall cost of the

piping system is determined by the combined specific piping and trenching costs.

20 25 32 40 50 65 80 100 125 150 200 250

Nominal pipe diameter DN

100

200

300

400

Specific trench costs ˆctrench [€/m]

Open field Street

Figure 2.16: Specific trench costs bases on Reference [70] and converted to the base year

2022 using conversion rates provided in Table A.3.

The design of DHN pipes is a technical-economic issue. Generously dimensioned pipes

lead to lower pumping costs due to reduced pressure losses but increased investment costs,

and vice versa [52]. Additionally, technical constraints limit the maximum permissible flow

velocity in a pipe, as excessive flow velocities can cause cavitation and noise development

[70]. Furthermore, high flow velocities lead to high pressure losses that must be compen-

sated by pumps, which can cause the allowed operating pressure to be exceeded, even in

short networks. Nussbaumer et al. [68] have shown that DHNs achieve the lowest costs

when the smallest technically feasible pipe diameter is selected. Figure 2.17 shows the

maximum feasible flow velocity with respect to the inner pipe diameter for two available

sources. It can be observed that allowed flow velocities increase as the inner pipe diameter

increases. According to ÖKL information, the flow velocity of connection lines should be

lower than transmission lines, mainly to reduce noise emissions [65]. Additionally, it is

common to design piping systems based on a nominal pressure difference Δ𝑝n, which is

defined as the pressure loss per meter of pipe length. In recent years, a nominal pressure

loss ranging from 150...250Pa/m has become established as economically advantageous

[52].

0 50 100 150 200 250 300 350 400

Inner pipe diameter din [mm]

Maximum feasible

flow velocity vmax [m/s]

Swedish DHA

OKL information

(Conection Lines)

100 Pa/m

200 Pa/m

300 Pa/m

Figure 2.17: Maximum feasible flow velocity as function of the inner pipe diameter based

on Reference [70].

2.4. District Heating Distribution Networks

2.4.5 Pressure System

The heat distribution of a DHN is realized by a closed hydraulic system, which is adapted

to temperature changes. A pump is increasing the differential pressure of the system,

which is reduced by pressure losses of the flowing heat transfer fluid through the network.

A pressure maintenance system ensures a constant resting pressure in the system and

compensates temperature-related volume fluctuations of the heat transfer fluid.

The pressure difference of a hydraulic system with constant inner pipe diameter 𝑑pip,in

can be expressed by Bernoulli’s equation extended by a pressure loss term [83] (see Equa-

tion (2.20)). Here, 𝜚is the density of the heat transfer fluid, 𝑔the gravitational constant,

Δℎ the height difference, 𝑣the average flow velocity, 𝐿the pipes length, 𝑓∗the pipe

friction factor, and ∑𝑖𝜉𝑖the overall pressure loss coeﬀicient of all fixtures. Term 1

represents the geodetic pressure difference, term 2the pressure loss due to pipe friction,

and term 3the pressure loss due to fixtures. The latter may be complicated to estimate,

especially in the early planning phases, because detailed knowledge about the selected

fixtures is needed. According to experience, the pressure losses of fixtures can be esti-

mated by 10...20%of the pressure losses of the straight pipe [70]. The geodetic pressure

difference can be neglected when designing the pump of a closed system because negative

and positive differences in height compensate [73]. When considering the total pressure of

the system, e.g., when determining the total pressure in the system, the geodetic pressure

difference must be taken into account.

Δ𝑝=𝜚𝑔Δℎ

⏟

1+𝜚

2𝑣2𝑓∗𝐿

𝑑pip,in

⏟⏟⏟⏟⏟

+𝜚

2𝑣2∑

𝑖𝜉𝑖

⏟⏟⏟⏟⏟

(2.20)

Bellos et al. [84] presented a formulation of the pipe friction factor that is valid for all

flow regimes (see Equation (2.21)).

𝑓∗=(64

Re)𝐾1[0.75ln (Re

5.37)]2𝐾2(𝐾1−1)[0.88ln(3.41𝑑in

𝑟)]2(𝐾1−1)(1−𝐾2)(2.21)

Where

𝐾1=1

1+(Re

2712)8.4 (2.22)

𝐾2=1

1+( Re

150𝑑in

𝑟)1.8.(2.23)

According to Equation (2.20), the pressure distribution in a DHN depends on a pres-

sure increase generated by the pumping station, pressure losses, and geodetic pressure

differences in the system. Figure 2.18 displays the pressure distribution of a DHN. The

pressure profile can be separated into a profile due to differential pressures and a pro-

file due to the altitude (see Figure 2.18a). The highest pressure difference between the

supply and return occurs at the pumping station Δ𝑝pump and continuously decreases due

to pressure losses until the minimum differential pressure Δ𝑝crit is reached at the critical

consumer. The critical consumer is the consumer in a DHN characterized by the lowest

differential pressure between supply and return due to pressure losses. The minimum pos-

sible pressure difference between supply and return is defined by the minimum differential

pressure at which the substation of the critical consumer can operate suﬀiciently. Typical

values range from 0.4...1.0bar [74, 75, 70]. The pressure loss in the supply pipe is de-

noted as Δ𝑝loss,s, and the pressure loss in the return pipe is Δ𝑝loss,r. In large DHNs, great

2. Theoretical and Conceptual Framework

Pressure

Network Length

Altitude

(a) Differential pressure and altitude profile.

Pressure

Network length

(b) Absolute pressure profile.

Figure 2.18: Characteristic pressure distribution of a DHN.

pressure losses can occur, leading to high differential pressures of the pumping stations,

which may exceed the nominal pressure (PN) of the piping system. In this case, the total

pressure increase can be distributed among several pumping stations located within the

DHN. The absolute pressure level is defined by the resting pressure 𝑝𝑅, which is usually

maintained by the pressure maintenance system. The differential pressure profile is super-

imposed with the pressure profile due to the geodetic difference in altitude to form the

absolute pressure profile shown in Figure 2.18b. The absolute pressure of a DHN must

always be within upper and lower limits to ensure secure operation. The maximum total

pressure resistance of a DHN is defined by the PN value of the selected piping system and

represents the maximum absolute pressure limit that can occur in a system for a short

period. It is usually defined as the maximum incremental pressure (MIP). The maximum

operating pressure (MOP) is smaller than the MIP and represents the maximum system

pressure under nominal conditions. The highest absolute pressure of a system 𝑝test occurs

during security tests performed after the pipe installation. The lower operating pressure

is defined by the minimum operating pressure (MinOP). The operating pressure must not

drop below the MinOP at any time to prevent the formation of evaporation and cavitation.

The MinOP is defined as the evaporating pressure 𝑝eva at maximum temperature plus a

safety margin. Typical values are 0.5...1.0bar. [70, 52]

From a hydraulic perspective, a district heating network can be characterized by the

interaction between the network’s distribution losses and the pump. The differential pres-

sure and volume flow through the pump result from the intersection of the pump and

network characteristics (see Figure 2.19). The network’s pressure loss coeﬀicient, denoted

as 𝜉dhn, is variable and depends on the position of the consumer control valves. When a

consumer’s heat request increases, the control valves open, and the pressure loss coeﬀicient

decreases, and vice versa. A change in 𝜉dhn alters the operational point.

Pumps can be designed as either controlled or uncontrolled, with uncontrolled pumps

typically being ineﬀicient and no longer in use [70]. When a pump is uncontrolled, the

operation point moves along the characteristic curve of the pumping station. In the case

of controlled pumps, the pump’s speed is adjusted to achieve a specified setpoint value

of the differential pressure. Two types of pressure control are possible: constant and

proportional, both of which are shown in Figure 2.19. For constant pressure control,

the pump’s differential pressure remains constant, while for proportional pressure control,

the differential pressure varies proportionally to the volume flow. Proportional pressure-

controlled pumps are generally more eﬀicient than constant pressure-controlled pumps.

2.4. District Heating Distribution Networks

Uncontrolled

Network

Pump

Figure 2.19: Characteristic differential pressure to volume flow curve of the network and

pump for uncontrolled, constant and proportional pressure differential control.

The electrical power consumption of a pump, 𝑃el,pump, can be calculated using Equa-

tion (2.24). Here, Δ𝑝pump refers to the pressure increase provided by the pump, 

𝑉pump

is the corresponding volume flow rate of the pump, and 𝜂pump is the combined hydraulic

and generator eﬀiciency of the pump. The annual electrical demand can be calculated by

integrating the electrical power consumption over the operational time 𝜏op using Equation

(2.25).

𝑃el,pump =

𝑉pump Δ𝑝pump

𝜂pump

(2.24)

𝑊el,pump,𝑎 =∫1a

𝜏=0h𝑃el,pump d𝜏(2.25)

The flow rate, pressure difference, and pump eﬀiciency typically vary depending on

the requested heat demand of the DHN and the selected control strategy of the pump.

Therefore, the integral in Equation (2.25) must be evaluated according to the given indi-

vidual load situation. In a planning process, this information is usually not known a priori.

For this reason, Nussbaumer et al. suggested a simpler approach to estimate the annual

power consumption [68]. They evaluated Equation (2.25) using the power consumption of

a network under nominal conditions and took the annual full load hours of the heat pro-

duction instead of the annual operation hours of the network. Furthermore, they replaced

the pump’s fluctuating eﬀiciency with its annual eﬀiciency 𝜁pump.

Thus, Equation (2.25) simplifies to Equation (2.26). Here, the nominal pressure differ-

ence of the pump refers to twice4the nominal pressure difference of the network and the

minimum pressure difference in each consumer substation Δ𝑝min,sst.

𝑊el,pump,a≈

𝑉pump Δ𝑝pump,n

𝜁pump =

𝑄n(2𝐿Δ𝑝n+Δ𝑝min,sst)

Δ𝜗n𝜁pump 𝜚𝑐p

(2.26)

In order to assess the error associated with the simplified approach presented in Equa-

tion (2.26) for estimating the pumping energy, an analysis of four representative network

control concepts within a DHN is provided in Section A.1.1, as outlined in the appendix.

The findings suggest that the estimation of annual pumping energy leads to a considerably

conservative result, making it highly advantageous for the planning phase.

2.4.6 Heat Losses

Heat losses in DHNs occur due to the average temperature of the DHN being higher than

the ambient temperature. The thermodynamic relations used to describe these heat losses

4For a two-level network with one supply and one return pipe.

2. Theoretical and Conceptual Framework

depend on the pipe installation situation. Currently, buried pipes are commonly used in

DHNs for heat distribution networks [70]. Therefore, in this section, we will only analyze

buried pipes.

If a two-level DHN is assumed, pipes can be installed as two buried single pipes or one

buried double pipe. Different thermodynamic considerations arise for each configuration.

For both pipe types, the installation situation leads to a transient three-dimensional heat

transfer problem. The inner pipe temperature, ambient temperature, and undisturbed

temperature of the soil are valid boundary conditions.

Numerical methods, such as finite element methods, can be used to solve such heat

transfer problems [80, 70]. However, the complexity of this consideration can be reduced

to a steady-state quasi-one-dimensional heat transfer problem because of the following

reasons:

1. The pipe insulation reduces the temperature difference between the inner pipe and

the environment by 80...90%, and the thickness of the pipe insulation is small com-

pared to the thickness of the surrounding soil. Hence, the temperature distribution

in the insulation can be assumed to be stationary [70].

2. The heat conduction in the flow direction is usually small and can be neglected [85].

3. The detailed operation of the DHN is usually not known at the design phase, and

temperature oscillations in the pipes have a short periodic time [70]. Therefore, the

temperature fluctuation in the pipes can be assumed to be constant.

4. The averaged values for ambient and soil temperatures can be used to estimate DHN

heat losses reasonably well [70].

Many studies have analyzed this steady-state heat transfer problem and derived pos-

sible solutions for engineering applications (see References [86, 87, 88, 89, 90, 73, 91]). A

widely used method [70, 88] that is recommended by the DIN EN 13941-1 [82] is the multi-

pole method suggested by Wallentén [87]. This method makes the following assumptions:

1. The convective thermal resistance between the heat transfer fluid and the inner pipe

surface is negligible compared to the thermal resistance of the pipe insulation and

can be neglected.

2. The temperature reduction of the heat transfer medium due to heat losses along the

network path are negligible.

3. The thermal resistance of the inner and outer pipes is negligible compared to the

thermal resistance of the insulation and can be neglected.

4. The method is only valid for two single pipes, and both pipes have the same dimen-

sions and are buried at the same depth. For one double pipe, both inner pipes must

have the same diameter.

Figure 2.20 presents the geometrical characteristics of a buried double pipe. The heat

loss of such pipes can be expressed by Equation (2.27) as suggested in [82], where therm

1denotes the thermal resistance of the pipe insulation, therm 2denotes the thermal

resistance of the soil, and therm 3refers to the interaction of both pipes. The variable

Δ𝜗mrepresents the average temperature difference between the nominal design tempera-

tures of the network and the ambient. Since the heat transfer problem was reduced to a

quasi-one-dimensional heat transfer problem, the average temperature difference between

2.4. District Heating Distribution Networks

the DHN and the environment can be evaluated using Equation (2.28). If the temperature

loss of the heat transfer medium is to be considered, the logarithmic temperature difference

is suitable for heat loss calculations. See Section A.1.4 for a more detailed description. To

account for the convective heat transfer coeﬀicient between the ground and the ambient,

which affects the heat transfer, the corrected depth of the pipes 𝐿dep,cor is suggested by [88]

and can be calculated using Equation (2.29). The convective heat transfer coeﬀicient 𝛼sur

in Equation (2.29) can be assumed to be 𝛼sur =14.6W/m2/K, which includes convection

and radiation of the ground surface to the ambient [88].



𝑄loss =4𝜋𝐿Δ𝜗m

𝜆ins ln(𝑑out

𝑑in )

⏟⏟⏟⏟⏟

1+1

𝜆soil ln(4𝐿dep,cor

𝑑out )

⏟⏟⏟⏟⏟⏟⏟

2+1

𝜆soil ln{[(2𝐿dep,cor

𝐿pip+𝑑out )2+1]0.5}

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(2.27)

Δ𝜗m=𝜗s,n+𝜗r,n

2−𝜗amb (2.28)

𝐿dep,cor =(𝐿dep +𝜆soil

𝛼sur )(2.29)

Figure 2.20: Geometric characterization for heat loss estimations of two single buried pipes

based on Reference [82]. Inner and outer diameters refer to the pipe insulation.

The annual heat loss of a network can be obtained by evaluating the integral of the

heat loss flow over the operating hours of the network, as described by Equation (2.30).

Since the heat loss is assumed to be in a steady-state, the heat loss flow is assumed to be

constant, thus enabling the integral to be solved.

𝑄loss,a=∫𝜏op

𝜏=0h

𝑄loss d𝜏= 

𝑄loss 𝜏op (2.30)

Equation (2.27) represents a series connection of multiple thermal resistances. Term

1donates the thermal resistance of the pipe insulation 𝑅1, term 2donates the thermal

resistance of the soil 𝑅2, and term 1donates the thermal resistance resulting from the

interaction of both pipes 𝑅3. In practical scenarios, it may be advantageous to simplify

Equation (2.50) by neglecting insignificant thermal resistance terms. To assess the impact

of each resistance, a study was conducted to analyze the thermal resistances 𝑅1–𝑅3

concerning their input parameters. The detailed analysis of each thermal resistance is

provided in Section A.1.2 of the appendix. The results indicate that the thermal resis-

tance 𝑅1dominates the overall resistance, accounting for a range of 89.2...94.8%of the

overall thermal resistance. The contribution of 𝑅2ranges between 4.3...8.7%, while 𝑅3

contributes between 0.9...2.1%of the overall thermal resistance depending on the nominal

diameter.

2. Theoretical and Conceptual Framework

2.4.7 Nominal Temperature Difference

The determination of the nominal supply temperature 𝜗s,nand the nominal temperature

difference between supply and return Δ𝜗nare critical parameters for the design of a DHN.

These parameters are responsible for defining the average temperature of the system, which

directly affects the heat losses. Moreover, the nominal temperature difference is a crucial

value when estimating the pipe diameters. According to the first law of thermodynamics,

the nominal temperature difference determines the thermal power that can be transported

with a given volume flow rate, as demonstrated in Equation (2.31). Thus, a decrease in

the temperature difference results in larger pipe diameters, since the maximum possible

flow velocity is limited, as shown in Figure 2.17.



𝑄n= 𝑚n𝑐pΔ𝜗n=

𝑉n𝜚𝑐pΔ𝜗n=𝑣n𝜋

4𝑑2

in 𝜚𝑐pΔ𝜗n(2.31)

Therefore, in order to optimize network design, the network designer should aim to

achieve low values of the supply temperature and high values of the nominal design tem-

perature difference. However, these values cannot be independently set due to their in-

terdependence. Specifically, the lowest possible nominal supply temperature of a DHS is

determined by the required supply temperature level of consumers with the highest sup-

ply temperature demand, while the nominal return temperature 𝜗r,nis determined by the

installed heating systems of the consumers. Thus, these two values are inherently linked.

20 40 60 80 100 120 140 160

Nominal supply temperature ϑs,n[°C]

100

Nominal temperature

difference ∆ϑn[K]

Case study data (n=44)

Case study regression

Case study extrapolation

Adapted extrapolation

Figure 2.21: Nominal temperature difference as function of the nominal supply tempera-

ture. Case study data was derived from Reference [30].

To evaluate the nominal supply and return temperature of DHSs, a case study was

conducted, which is described in detail in Reference [30]. The study presents a linear cor-

relation between the nominal supply temperature and the nominal temperature difference

of 44 DHNs in Switzerland, as shown in Figure 2.21. However, the data was only available

for nominal supply temperatures within the range of 𝜗s,n=70...170°C. For lower nominal

supply temperatures an additional description is required. When a linear extrapolation is

applied to the case study regression, negative values for the nominal temperature differ-

ence were obtained for low values of the nominal supply temperature, which is physically

impossible. Therefore, an adjusted extrapolation was utilized, which involved assuming

an extreme supply scenario where the nominal supply temperature and the nominal re-

turn temperature are equal to the comfort temperature of a heated building. Specifically,

Δ𝜗n=0K at 𝜗s,n=20°C. It should be noted that this theoretical assumption is unattain-

able in reality due to the minimum temperature difference required for heat transfer. By

making this assumption, the correlation between the nominal temperature difference and

the nominal supply temperature can be expressed using Equation (2.32). Additionally,

it should be noted that this approach is based on the assumption that the nominal tem-

perature difference, and consequently the maximum mass flow rate, is achieved when the

2.5. Planning Phases of District Heating Systems

nominal supply temperature is reached. While this assumption is true for most scenarios,

its validity depends on the chosen operation mode of the supply temperature, as described

in Section 2.3.4.

Δ𝜗n={0.726𝜗s,n−30.595°C if 𝜗s,n≥70°C

0.420𝜗s,n−8.404°C if 𝜗s,n<70°C(2.32)

2.5 Planning Phases of District Heating Systems

As with other sectors of the energy industry, the economic feasibility of DHSs is critical to

successful project implementation. These systems face direct competition from decentral-

ized heat supply systems, and their success cannot be guaranteed by simply replicating

solutions from other areas due to the uniqueness of each supply region. Costs can vary

greatly within a single supply area, highlighting the importance of careful planning to

ensure successful DHS implementation. However, there are certain criteria that can guide

the estimation of a successful supply solution during the planning process. The DHS com-

prises a heat generation and heat distribution system, with this thesis focusing on the

latter aspect, and therefore excluding heat generation planning from consideration in this

section.

According to Reference [70], the planning process for a DHS can be divided into six

phases, of which four fall under the planning phase and two under the operational phase.

These phases are illustrated in Figure 2.22. As the project progresses, the level of detail and

project costs increase, with the implementation and acceptance phase typically creating

the highest costs.

1. Preliminary

study

2. Design

planning

3. Planning,

tendering and

awarding

6. Operation

and management

Implementation

and acceptance

5. Operation

optimization

Planning phase Operational phase

Project costs

Cancellation?

0.1 - 0.2 % 0.5 - 1.5 % 4.5 - 5.5 % 92 - 95 % 100 %

Cancellation?

Scope of this Thesis

Figure 2.22: Planning phases to realize the heat distribution of a DHS based on References

[70, 52].

During phase one, the preliminary study establishes the basis for the project’s further

development. The aim is to acquire and process data and provide statements on the

feasibility, risks, and benefits of the project. Tasks include identifying the supply area and

key consumers, estimating heat and temperature demand, identifying load characteristics,

and selecting the typology and type of the DHN and piping system. Economic feasibility

is also roughly estimated. At the end of this phase, a decision is made whether the project

is within budget limits, or if the project should be cancelled. Project costs at this point

are just 0.1...0.2%of total costs, justifying project cancellation if necessary. Accurate

estimates at this point will reduce risks in further project development [70, 52].

Phase two involves the concretization of estimations made in phase one and estab-

lishing several economically feasible scenarios. The consumer database’s level of detail is

increased, and the heat supply area is specified. A suitable piping system is selected and

designed based on these boundary conditions. At the end of this phase, a second economic

evaluation is conducted and compared to budget planning. Project costs at this point are

2. Theoretical and Conceptual Framework

between 0.5...1.5%of total costs, making cancellation of the project acceptable, but not

ideal [70, 52].

The aim of phase three is to finalize planning so that implementation can start. The

main tasks include finalizing the DHN and substation designs, creating plans ready for im-

plementation, preparing tenders and obtaining offers, realizing the awarding, and finalizing

economic considerations [70]. By the end of this phase, about 4.5...5.5%of total project

costs have been reached [52]. Cancellation at this point is associated with significant costs

and should only be done if really necessary.

The implementation of the DHN occurs in phase four based on detailed planning

documents. At the end of this phase, the system undergoes functionality testing, and the

DHN is handed over to the owner. Usually, 92...95%of total project costs are spent by

the end of this phase [52].

Eﬀicient operation of the DHS according to planning specifications requires operational

optimization. Correct planning and implementation do not necessarily guarantee eﬀicient

operation, especially for complex systems. Therefore, systematic operational optimization

is necessary to ensure eﬀicient operation. Optimization is usually done within the first two

years after commissioning, and data collection should be done for at least one year. By

the end of phase five, the DHS is planned, implemented, and optimized, with total project

costs incurred [70, 52].

Finally, phase six involves the operative operation and management of the DHS, in-

cluding regular operation, repair, and maintenance [70].

2.6 Economic Considerations

Besides technical considerations, the feasibility of supplying an area with district heat-

ing mainly depends on economics. DHS compete with other energy systems, which are

primarily decentralized heating systems in this context [9]. Compared to decentralized so-

lutions, DHSs require investments in the necessary heat distribution infrastructure, which

must also be maintained [10]. Additionally, heat and pressure losses occur due to heat

distribution, which need to be compensated [10].

The VDI guideline 2067 [92] describes the costs of building installations, which can

also be applied to district heating systems, according to Reference [70]. Typical cost

components are capital-related, demand-related, operation-related, and other costs [92,

70]. In this context, capital-related costs include the costs of the investment of the district

heating system, including interest on the used capital and including costs for continuous

renovation. Demand-related costs are cost components that usually scale according to

the produced or consumed energy. These costs include, for example, costs of fuels or

other energy carriers, consumption of auxiliary energy, and costs of additional supplies.

Operation-related costs include costs required for operation and maintenance. Other costs

refer to costs that are not covered by the given cost components, such as taxes, insurances,

profits, and administration costs.

Specific costs in the form of the levelized costs of heat (LCOH) allow describing the

costs of district heating systems. In general, the LCOH is a measure of the average costs

for heat generation and distribution over the entire lifetime of a district heating system.

The LCOH represents the average price that would have to be paid by consumers to repay

the investor and/or operator for all expenses. The LCOH can be expressed in a general

representation regarding Equation (2.33).

LCOH =𝑐= Sum of costs over lifetime

Sum of thermal energy sold to the consumers (2.33)

2.6. Economic Considerations

According to the literature, the LCOH of a DHS 𝑐DHS can be separated into heat

generation, heat transmission, and heat distribution costs [18, 93, 13, 38, 94] (see Equation

(2.34)). The heat generation costs 𝑐hg refer to the costs related to the production of heat

by the heat generator. Heat transmission costs 𝑐trm represent the expenses of transporting

heat from the heat generator to the distribution network, which is mainly relevant if

the heat generator and the distribution network are significantly separated. The heat

distribution costs 𝑐dst describe the costs that are required to distribute heat within a

supply area. The costs of heat transmission and heat distribution refer to the costs of the

DHN, which is why they can be considered as total heat distribution costs 𝑐dst,tot.

𝑐DHS =𝑐hg +𝑐trm +𝑐dst =𝑐hg +𝑐dst,tot (2.34)

DHSs can be considered economically feasible if the heat production and overall heat

distribution costs are lower than the costs of competing technologies [44]. Otherwise,

decentralized heating systems are advantageous and should be considered as a heat supply

technology. In general, it is very diﬀicult to determine the share of heat generation and

heat distribution costs, since they are mainly project-related and depend on individual

site conditions [9]. Oppermann et al. [52] stated that the ratio of heat distribution to heat

generation costs falls within a wide range of 𝑐dst,tot/𝑐hg =0.176...1.17, which underscores

these diﬀiculties.

2.6.1 Heat Generation Costs

According to Reference [8], the LCOH of a heat generator can be expressed by Equation

(2.35). Here, 𝑐cap,hg represents the specific capital-related costs of heat generation, 𝑐f,hg

represents the specific fuel costs, and 𝑐om,hg represents the specific O&M costs of the heat

generator, which can be separated into fixed and variable O&M costs. The fuel costs

represent demand-related costs, and both shares of the operation and maintenance cost

represent operation-related costs.

𝑐hg =𝑐cap,hg +𝑐f,hg +𝑐om,hg (2.35)

The specific capital costs of the heat generator can be expressed using Equation (2.36)

[8, 18, 70]. In this context, 𝑐cap,hg represents the specific investment costs per installed

capacity, 

𝑄hg,nrepresents the installed nominal capacity of the heat generator, and 𝑎

represents the annuity factor. The annuity factor is an expression that allows the present

value of a series of equal periodic payments that are expected to be received or paid out

over a fixed period of time to be calculated. It takes into account the interest rate IR of

the capital invested and the duration of the investment 𝜏inv (see Equation (2.37)). The

specific fuel costs are defined as the total fuel costs related to the annual heat sold to the

customer, as given by Equation (2.38). Here, 𝑄hg,arepresents the annual heat produced,

𝜁th represents the annual eﬀiciency of the heat generator, and 𝑐∗frepresents the specific

fuel price. The O&M costs can be expressed as a fixed and a variable share (see Equation

(2.39)). Here, 𝑐om,fix,hg represents the specific fixed O&M costs per installed heat generator

2. Theoretical and Conceptual Framework

capacity, and 𝑐om,var,hg represents the specific variable O&M costs per produced heat.

𝑐cap,hg =1

𝑄con,a𝑐inv,hg 

𝑄hg,n𝑎(2.36)

𝑎= (IR +1)𝜏inv IR

(IR +1)𝜏inv −1 (2.37)

𝑐f=1

𝑄con,a𝑄hg,a

𝜁th 𝑐∗f(2.38)

𝑐om,hg =1

𝑄con,a( 𝑐om,fix,hg 

𝑄hg,n+ 𝑐om,var,hg 𝑄hg,a)(2.39)

Please refer to Section 2.3.5 for a detailed description of specific investment costs of

different types of heat generators and typical fuel prices.

The O&M costs of a heat generator can be divided into a fixed and variable share.

Fixed O&M costs are typically paid per year and depend on the installed capacity. These

costs include all expenses that are independent of the operation hours, such as adminis-

tration, insurances, network or system charges, and property tax [8]. On the other hand,

variable O&M costs depend on the operation hours of the plant and include consumption

of auxiliary energy and materials, treatment of disposals, spare parts, and output-related

repair and maintenance. Table A.6 in the appendix provides an overview of typical values

of economic and technical parameters of different heat generators, including O&M costs.

2.6.2 Heat Distribution Costs

In contrast to heat generation costs, which are characterized by economy-of-scale, heat

distribution costs are characterized by economy-of-scope [20]. This means that the heat

distribution costs of DHNs are mainly project-related [9] and depend highly on the indi-

vidual use case. This makes it diﬀicult to assess heat distribution costs before a detailed

analysis of the individual project has been carried out.

Generally, the overall heat distribution costs of a DHS 𝑐dst,tot consist of capital, pump-

ing, heat loss, and O&M costs [44, 18, 68], which can be formulated according to Equation

(2.40). Here, 𝑐cap,dst refers to the specific capital costs of the distribution network, 𝑐pump,dst

to the specific pumping costs, 𝑐loss,dst to the specific heat loss costs, and 𝑐om,dst to the spe-

cific O&M costs of the heat distribution system.

𝑐dst =𝑐cap,dst +𝑐pump,dst +𝑐loss,dst +𝑐om,dst (2.40)

Capital Costs

The capital costs of a DHNs mainly consists of capital costs related to the piping system

𝑐cap,pip, the pumping stations 𝑐cap,pump, and the consumer substations 𝑐cap,sst [10] (see

Equation (2.41)).

𝑐cap,dst =𝑐cap,pip +𝑐cap,pump +𝑐cap,sst (2.41)

𝑐cap,pip =𝑎𝐶inv,pip

𝑄con,a=𝑎𝐿(𝐼0+𝐼1𝑑in)



𝑄con 𝜏full =𝑎(𝐼0+𝐼1𝑑in)

𝑞lin

(2.42)

𝑐cap,pump =𝑎𝐶inv,pump

𝑄con,a=𝑎

𝑄dhn,n𝑐inv,pump 𝑛pump

𝑄con,a

(2.43)

𝑐cap,sst =𝑎𝐶inv,sst

𝑄con,a=𝑎∑𝑛con

𝑖=1 

𝑄con,n,𝑖 𝑐inv,sst,𝑖

𝑄con,a

(2.44)

2.6. Economic Considerations

The capital costs of the piping system can be calculated according to Equation (2.42).

Hence, they mainly depend on the investment costs of the piping network 𝐶inv,dst, the

annuity factor 𝑎, and the annual heat sold to the customer 𝑄con,a. The investment costs

of the distribution network depend on the pipe trench length 𝐿, the inner pipe diameter

𝑑in, and the linear piping cost coeﬀicients 𝐼0and 𝐼1. The annual heat sold to the customer

can be expressed by the overall nominal consumer power 

𝑄con and the full load hours 𝜏full.

The linear piping coeﬀicients 𝐼0and 𝐼1represent a linear relation between the specific

pipe investment costs and the inner pipe diameter. Figure 2.23 shows an example of possi-

ble piping investment costs related to the inner pipe diameter for underground installation

of KMR pipes. Here, a nearly linear relationship between the internal pipe diameter and

the specific investment costs can be observed. Only at smaller diameters (𝑑in <0.05m)

certain deviations from the linear shape are noticeable.

0.05 0.10 0.15 0.20 0.25

Inner pipe diameter din [m]

500

1000

1500

2000

Specific pipe investment costs

Cinv/L [€/m]

Street

Street Regression y= 6246.7/m2

·di+ 219.4/m

Open Field

Open Field Regression y= 5732.2/m2

·di+ 144.3/m

Figure 2.23: Linear relationship between inner pipe diameter and specific pipe investment

costs for underground installation of KMR pipes with insulation class 2. The data is based

on prices from [68] and has been converted to the base year 2022 using conversion rates

provided in Table A.3.

The inner diameter of a pipe can be determined using Equation (2.45), where 𝑓∗

represents the pipe friction factor, Δ𝜗nrepresents the nominal temperature difference

between supply and return, 𝜚represents the density of the heat transfer fluid, 𝑐prepresents

the specific heat capacity of the heat transfer fluid, and Δ𝑝nrepresents the nominal design

pressure loss of the DHN. For a detailed derivation of this equation, refer to Appendix

A.1.3. 𝑑in =5

√8

𝑄2

n𝑓∗

Δ𝜗2

n𝜋2𝜚𝑐p2Δ𝑝n

(2.45)

Since the friction factor depends on the pipe diameter (see Equation (2.21)), Equation

(2.45) can only be solved iteratively if no different formulation of the friction factor is used.

Merging Equations (2.42) and (2.45) leads to Equation (2.46), which provides a general

formulation of the capital costs of a DHN. This Equation indicates a root function behavior

of the network capital costs if all remaining parameters, except the network length 𝐿, are

assumed to be constant.

𝑐cap,pip =𝑎

𝑞lin (𝐼0+𝐼15

√8𝑞2

lin 𝐿2𝑓∗

Δ𝜗2

n𝜋2𝜚𝑐p2Δ𝑝n𝜏2

full )(2.46)

The capital costs of the pumping station can be calculated according to Equation (2.43).

Here, the investment costs of the pumping station 𝐶inv,pump can be expressed as the

nominal heat capacity of the DHN 

𝑄dhn,nmultiplied by the specific investment costs of

2. Theoretical and Conceptual Framework

a pumping station 𝑐inv,pump. Additionally, multiple pumping stations may be required if

high pressure losses occur. This is accounted for by the factor 𝑛pump. Table 2.7 provides

typical values of the specific investment costs of pumping stations. Depending on the size

of the DHN, two different specific investment costs are given.

Table 2.7: Overview of typical specific investment costs of pumping stations converted to

the base year 2022 using conversion rates provided in Table A.3 without VAT according

to Reference [8]. 

𝑄dhn,n[kWth]<1000 ≥1000

𝑐inv,pump [€/kWth]209–313 78–117

The capital costs of the consumer substations can be calculated according to Equation

(2.44). Here, the investment costs of the substations 𝐶inv,sst are composed of the investment

costs of all substations in the distribution area. The variable 

𝑄con,n,𝑖 corresponds to the

nominal connection capacity of the consumer 𝑖and 𝑐inv,sst,𝑖to the specific investment costs

of each substation. For a detailed overview of substation investment costs, see Section

2.2.4.

Pumping Costs

The power consumption of a DHS mainly depends on the electrical consumption for pump-

ing, which can be determined by the volume flow rate 

𝑉, the pressure difference provided

by the pump Δ𝑝pump, the annual operating hours of the network 𝜏op, the pumps eﬀiciency

(hydraulic and electric) 𝜂pump, and the control concept of the pump. A general formulation

of the pumping costs is shown in Equation (2.47).

𝑐pump,dst =𝑐el

𝑄con,a∫𝜏op

𝜏=0h



𝑉Δ𝑝pump

𝜂pump

d𝜏(2.47)

Using an approximation of the annual power consumption introduced in Section 2.4.5,

Equation (2.47) simplifies to Equation (2.48). Here, the nominal pressure difference of the

pump refers to double5the nominal pressure difference of the network and the minimum

pressure difference in each consumer substation Δ𝑝min,sst.

𝑐pump,dst =𝑐el Δ𝑝pump,n

Δ𝜗n𝜁pump 𝜚𝑐p=𝑐el 2𝐿Δ𝑝n+Δ𝑝min,sst

Δ𝜗n𝜁pump 𝜚𝑐p

(2.48)

Heat Loss Costs

The LCOH of heat losses occurring from the heat losses of the DHN pipes to the ambient



𝑄loss can be generally described by Equation (2.49), where 𝑐th refers to the specific heat

production costs for heat fed into the DHN, and 

𝑄loss is the heat loss rate.

𝑐loss,dst =𝑐th

𝑄con,a∫𝜏op

𝜏=0h

𝑄loss d𝜏(2.49)

Different descriptions of heat loss are possible depending on the installation situation.

For instance, if two buried pipes are to be investigated, Equations (2.27) – (2.29) can be

used to describe the heat losses. When the temperature difference Δ𝜗mis estimated using

5For a two-level network with one supply and one return pipe.

2.7. Pipe Distribution Configurations

an averaged and constant value (see Equation (2.28)), the LCOH for heat losses of two

buried pipes can be expressed by Equation (2.50). For a description of the corrected depth

𝐿dep,cor, refer to Equation (2.29).

𝑐loss,dst =4𝜋𝜏op 𝑐th Δ𝜗m

𝑞lin (1

𝜆ins ln(𝑑out

𝑑in )+ 1

𝜆soil ln(4𝐿dep,cor

𝑑out )+ 1

𝜆soil ln{[(2𝐿dep,cor

𝐿pip+𝑑out )2+1]0.5})

(2.50)

Operation and Maintenance Costs

The O&M costs of a DHN can be separated into fixed and variable components according

to Equation (2.51). Here, 𝑐om,fix,dst represents the specific O&M costs per installed heat

generator capacity, and 𝑐om,var,dst represents the specific O&M costs per produced heat.

Fixed O&M costs are independent of the annual transported heat, while variable O&M

costs scale with the annual transported heat of the DHN.

𝑐om,dst =1

𝑄con,a( 𝑐om,fix,dst 

𝑄th + 𝑐om,var,dst 𝑄th,a)(2.51)

Although O&M costs are often small in comparison to heat loss, pumping, and capital

costs, they should not be entirely neglected in the planning phase [68, 70]. The Danish

Energy Agency suggests using specific variable O&M costs of 𝑐om,var =0.15ct/kWh for

DHN and ignoring the fixed O&M cost [10]. If the heat losses of the DHN are neglected

when estimating the O&M costs, Equation (2.51) simplifies to Equation (2.52).

𝑐om,dst = 𝑐om,var,dst (2.52)

2.7 Pipe Distribution Configurations

According to Nussbaumer and Thalmann [95], a linear DHN can be designed using the

two basic pipe distribution principles SCC and DCC. Both concepts were introduced in

Section 1.2.4 and shown in Figure 1.3.

In terms of LCOH calculations presented in Section 2.6.2, the computational effort of

both configurations differs. The SCC configuration is characterized by a single-sized pipe

segment assuming all consumers are located at the end of the linear DHN, and Equations

(2.40) – (2.52) only need to be evaluated once for a particular network expansion. This

makes this consumer configuration suitable if a quick but rather coarse estimation of

the heat distribution costs is required. Furthermore, this configuration can be applied

if decentralized fed-in is considered at any location of the network, as the pipe diameter

is sized according to the accumulated nominal power of all consumers. This may be

advantageous if DHNs of the 4th or 5th generation are considered. Although this approach

does not represent the real diameter distribution with decentralized feed-in, it represents

an upper limit of the possible distribution costs that may occur.

In contrast to the SCC, a more realistic assumption without decentralized fed-in would

be a DCC, which is characterized by a gradually decreasing pipe diameter along the linear

network path. In this scenario, lower LCOH is expected because this configuration leads to

a smaller averaged pipe diameter than the SCC. Unlike the SCC, Equations (2.40) – (2.52)

need to be evaluated gradually along the linear network path at each extraction point,

which takes more computational effort but can be implemented easily using a computer

program.

2. Theoretical and Conceptual Framework

2.8 LCOH of Radial Structured DHNs

In addition to the simple linear path from the heat producer to consumers, DHNs can

have much more complex structures. As described in detail in Section 2.4.3, the topology

of real-world DHN applications can be categorized as radial or meshed networks. Radial

networks consist of linear segments branching off from a heat producer to distributed

consumers without any loops. This section will describe how the overall cost of a complex

radial network can be derived from the incremental costs of its linear segments.

An example of a radial DHN is shown in Figure 2.24. This network consists of one

producer and 36 consumers. The network structure is characterized by six sub-branches

branching off from one main branch at three nodes. In each sub-branch, three sub-sub-

branches branch off at three nodes, resulting in a repetitive network structure connecting

branches through nodes. For simplicity, all sub-sub-branches are neglected in the following,

resulting in seven identifiable branches that can be characterized by a column name (A or

B) and a row name (I, II, or III).

Radial connection

Linear connection

Consumer

Node

Heat generator

Pipes

Main branch

Sub-branch

A B

III

Figure 2.24: Example branched DHN consisting of a single heat generator, 36 consumer,

one main branch and six sub-branches.

Taking the given network structure into account, independent and dependent branches

can be identified. The volume flow rate of independent branches is not affected by any

changes made to the consumer heat capacity outside the considered branch. Dependent

branches, on the other hand, are influenced by changes made in the consumer heat capacity

outside the considered branch. The following considerations based on the selected example

network should clarify this concept.

A change in any consumer capacity in sub-branch A-I would lead directly to a change

in the nominal volume flow rate and therefore to a different piping design in this branch.

Simultaneously, the nominal volume flow rate of the main branch also changes. All re-

maining sub-branches are not affected by any changes made in sub-branch A-I. Thus,

the main branch is somewhat dependent on branch A-I, while the piping design of all

other sub-branches remains unchanged and can therefore be described as independent of

sub-branch A-I. This analysis can be repeated for each sub-branch, which shows that the

sub-branches are independent of each other, and the main branch is dependent on each

sub-branch. This principle is hereafter referred to as the principle of separation and can

be used to estimate the overall LCOH of complex DHNs by a combination of consisting

only linear network segments.

Using the principle of separation, the overall LCOH of the given example from Figure

2.24 can be separated into a dependent 𝑐dpt and an independent cost share 𝑐ipt regarding

Equation (2.53). See Figure 2.25 for a graphical explanation.

2.8. LCOH of Radial Structured DHNs

𝑐dhn =𝑐ipt +𝑐dpt (2.53)

Separation +

Independent branches Dependent

branch

Radial DHN

Figure 2.25: Graphical explanation of the principle of separation.

The individual LCOH of each branch can be calculated according to Equations (2.40)–

(2.52) in Section 2.6.2. Therefore, the LCOH of all liner segments is assumed to be known.

The LCOH of a branch is the ratio of the annual costs 𝐶arelated to the annual heat

demand of all consumers in that branch 𝑄con, according to Equation (2.54). The overall

LCOH of the given example DHN in Figure 2.24 can be expressed by the sum of the

costs of all branches divided by the annual overall heat demand of all consumers 𝑄tot,

according to Equation (2.55). Thus, the total LCOH can be divided into an independent

and dependent share 𝑐ipt and 𝑐dpt. The dependent share refers to the LCOH of the main

branch, and the independent share can be written as Equation (2.56). Thus, the overall

LCOH of all independent branches can be derived from the partial LCOH of each branch,

weighted by the partial heat demand of each sub-branch in relation to the overall heat

demand.

𝑐= 𝐶

𝑄con

(2.54)

𝑐tot =𝐶A−I+𝐶B−I+𝐶A−II +𝐶B−II +𝐶B−III +𝐶A−III

𝑄tot

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑐ipt

+𝐶main

𝑄tot

⏟

𝑐dpt

(2.55)

𝑐ipt =𝑐A−I𝑄A−I

𝑄tot +𝑐B−I𝑄B−I

𝑄tot +⋯+𝑐B−III 𝑄B−III

𝑄tot =𝑛ipt

∑

𝑖=1 𝑄𝑖

𝑄tot 𝑐𝑖(2.56)

The previous example shows that the total costs of a complex radial network can be

calculated, given that the costs of the sub-branches are known. Therefore, this thesis will

focus solely on linear distribution networks.

3 Detailed Simulation Model

This chapter presents a model that estimates the heat distribution costs of a linear DHN

based on a detailed simulation. The simulation model is associated with the detailed

network analysis approach introduced in Section 1.2.2. This approach enables the accurate

calculation of the heat distribution costs of a linear DHN. The developed model and the

underlying algorithm are described in detail in Section 3.1. All investigations carried out

using this model are introduced in Section 3.2, including a validation study, a default

model analysis, an one factor at a time (OFAT) parameter study, and a Monte Carlo

parameter study. The validation study presented in Section 3.3 assesses the accuracy of

the developed model in comparison to data found in the literature. The default model

analysis presented in Section 3.4 carries out a general analysis of the heat distribution costs

according to a set of defined default input parameters. Section 3.5 analyzes how a change

in a single input parameter affects the heat distribution costs. Section 3.6 investigates the

effect of changing several input parameters on the heat distribution costs and presents a

regression model to estimate the heat distribution costs based on a set of network design

parameters.

3.1 Model Design

This section presents a detailed model for estimating the specific heat distribution costs

(LCOH) of a linear DHN. The model, shown in Figure 3.1a, consists of a single heat

generator that supplies multiple heat consumers through a linear DHN. The DHN is

designed as a two-pipe system, comprising a supply pipe and a return pipe. The network’s

topology is characterized by a main branch to which consumers are connected via nodes.

Each consumer can be defined by a linear and radial position.

The linear position 𝑖indicates the location of a consumer’s corresponding node along

the main branch. Here, 𝑖=1corresponds to the first node adjacent to the heat generator,

and 𝑖=𝑛lin corresponds to the last node. The distance from the heat generator to a

node is represented by the length of the main branch 𝐿bra. The main branch is divided

by the nodes into 𝑛lin individual segments, which lengths are donated as Δ𝐿bra,𝑖. The

heat generator is located at 𝐿bra =0m, and the consumers farthest away from the heat

generator are located at 𝐿bra =𝐿bra,tot.

The number of consumers connected to the main branch at each node is described by

the number of radial connections 𝑛rad. The variable 𝑗denotes the radial location of each

consumer. In the given example (see Figure 3.1), 𝑛rad =2, whereas 𝑗=1represents the

consumers above and 𝑗=2represents the consumers below the main branch. The length

of each connection is denoted as Δ𝐿con,𝑖,𝑗. Furthermore, each consumer is characterized

by its nominal power 

𝑄con,n,𝑖,𝑗 and its corresponding annual full load hours 𝜏full,con,n,𝑖,𝑗.

An overview of additional model parameters is given in Table 3.1. How these parameters

are affecting the model outcome will be explained further below.

Table 3.1: Overview of technical and economical model parameter of the detailed model.

Technical parameters Economical parameters

𝑛lin;𝑛rad;𝑟;𝜗s,n;Δ𝜗n;𝜗amb;𝜁pump;

Δ𝑝n;𝑝R;𝜉n;Δ𝑝min,sst;𝑝max;𝜏op;𝑆𝐹;

𝜆ins;𝜆soil;𝐿dep;𝐼𝐶 𝐼0;𝐼1;𝐼𝑅;𝜏inv;𝑐OM,var,dst;

𝑐OM,fix,dst;𝑐∗

el;𝑐th;𝑐inv,pump;𝑐inv,sst

3. Detailed Simulation Model

Main branch

Connection Consumer

parameters:

Legend:

Consumer

Node

Pipe

Heat generator

1 2 30 4

Segment

(a) Toplogical structure.

DCC:SCC:

Segments i*

i = 1

i = 2

i = nlin

1 2 30 4

4..

1 2 30 4

i = 3

i*=1 i*=2 i*=3 i*=4 i*=i

Segments i*

i = 1

i = 2

i = nlin

i = 3

i*=1 i*=2 i*=3 i*=4 i*=i

(b) Main branch for several network expansions.

Figure 3.1: Model structure and graphical explanation of the corresponding linear DHN

of the detailed simulation model. Top view: topological structure of heat generator, the

DHN, and all consumers. Bottom view: size of the main branch for several network

expansion.

As discussed in Section 1.2.4, Nussbaumer et al. have observed that the expansion

of a linear DHN has an impact on the specific heat distribution costs. Consequently, the

focus of this thesis is on the expansion of the network in the linear direction. Figure 3.1b

provides a graphical representation of the network expansion, illustrating the length and

diameter variations of a pipe in the main branch for different network expansions and

consumer configurations (SCC or DCC).

The linear network can be characterized by different linear positions denoted as 𝑖.

Theoretically, it is possible to construct a linear network for each linear position, expanding

it from the heat generator until the corresponding linear position. In this context, it is

assumed that all consumers located between the heat generator and the corresponding

linear position are connected, while all remaining consumers are not connected to the DHN.

To illustrate this, a network that is expanded up to linear position 𝑖=3is considered. In

a linear network expansion up to this position, all consumers at linear positions 𝑖=1...3

are supplied by the DHN, while consumers at positions 𝑖=4..𝑛lin are not connected to the

DHN. This process can be repeated for each linear position, resulting in as many network

expansions as there are linear positions. In addition to that, the main branch of each of

the resulting linear networks can be separated into several segments, which are denoted

by the variable 𝑖∗.

Since longer network expansions involve supplying heat to more consumers, the overall

nominal power for which the network is designed increases, leading to larger pipe diameters

in the main branch. This effect is shown in Figure 3.1b for different network expansions

and both consumer configurations SCC and a DCC. In the case of a SCC, the diameter

reduction along the main branch is not considered, as this network configuration assumes

that all consumers are located at the far end of the network. In contrast, the network

3.1. Model Design

expansion of the DCC takes into account a reduction in pipe diameter along the main

branch.

The methodology employed to calculate the heat distribution costs of the linear district

heating network can be described by the six-step program flow chart presented in Figure

3.2. In the first step, all necessary constants are computed. Subsequently, the mass flow

rates of the consumers are calculated in the second step. The third step involves determin-

ing the mass flow rates and transportable thermal power at each network expansion and

in each segment, which forms the basis of the pipe design in step four. After designing

all the pipes, the entire network is computed in step five, which includes the calculation

of heat and pressure losses, as well as economic evaluations. The results are presented

through plotting and exporting in the final step six.

Model parameter

1. Get constants

2. Get consumer mass ﬂow

3. Get main branch parameters

4. Design consumer and branch pipes

5. Calculate network

6. Export and plot results

Continuous or

manufacturer's

speciﬁcations?

All data

Start

End

Figure 3.2: Program flow chart of the detailed simulation model. The program flow is

illustrated by the solid lines. For each step, several model parameters are required which

is illustrated by the dashed lines.

The initial step of the model is related to setting up the required model parameters,

which are summarized in Table 3.1. Additionally, the user can decide whether the nom-

inal temperature difference Δ𝜗nshould be an independent parameter or if it should be

determined based on the nominal supply temperature 𝜗s,nusing Equation (2.32).

3. Detailed Simulation Model

The first step of the simulation involves computing all necessary constants, which

include the specific heat capacity at constant pressure 𝑐p, the density 𝜚, and the dynamic

viscosity 𝜇of the heat transfer fluid. These parameters are derived from a reference

temperature 𝜗ref and a reference pressure 𝑝ref using the IAPWS-IF97-formulation [96].

The reference pressure is the resting pressure 𝑝R, and the reference temperature is the

average nominal supply and return temperature of the network, computed using Equation

(3.1). Additionally, the annuity 𝑎is a constant that is computed using Equation (2.37).

𝜗ref =𝜗s,n−Δ𝜗n

2(3.1)

In the second step, the nominal mass flow rate of each consumer is computed using

Equation (3.2). It is important to note that each consumer is connected to the main branch

through a single connection pipe, so the mass flow rate of the consumer is equivalent to

that of the connection pipe.

𝑚con,n,𝑖,𝑗 =

𝑄con,n,𝑖,𝑗

𝑐pΔ𝜗n

(3.2)

In the third step, the mass flow rates and nominal power of each network segment of the

main branch are calculated for all possible network expansions. The network expansion is

performed in a linear direction, starting at 𝑖=1and expanding until 𝑖=𝑛lin as illustrated

in Figure 3.1b. As the value of 𝑖increases, more consumers are connected to the linear

DHN, which affects the pipe design of the branch as a whole. It should be noted that heat

losses are neglected at this stage, but are taken into account when calculating the entire

network in step five. At this stage, a distinction is made between the SCC and DCC of the

main branch. This is because the nominal mass flow rate and the thermal power at a given

network expansion are the same in each segment at the SCC, but differ at the DCC. The

thermal power of each branch segment at the SCC is calculated using Equation (3.3), and

the corresponding mass flow rate is calculated using Equation (3.4). It should be noted

that the counting variable 𝑖corresponds to the network expansion, while 𝑖∗corresponds

to the segments of the expansion 𝑖. Additionally, the function SF (𝑛rad,𝑖)represents the

simultaneity factor according to Equation (2.7), which depends on the total number of

consumers.



𝑄bra,n,scc,𝑖 =SF(𝑛rad,𝑖)𝑖

∑

𝑖∗=1

𝑛rad

∑

𝑗=1 

𝑄con,n,𝑖∗,𝑗 (3.3)

𝑚bra,n,scc,𝑖 =

𝑄bra,n,scc,𝑖

𝑐pΔ𝜗n

(3.4)

The power of each branch segment 𝑖∗at a certain network expansion 𝑖at DCC can be

calculated using Equation (3.5), and the corresponding nominal mass flow rate can be

calculated using Equation (3.6). This approach assumes that the initial pipe segment

receives the full thermal power, which gradually reduces at increasing network sections.



𝑄bra,n,dcc,𝑖,𝑖∗=SF(𝑛rad,𝑖)( 

𝑄bra,n,scc,𝑖−𝑖∗−1

∑

𝑘=1

𝑛rad

∑

𝑗

𝑄con,n,𝑘,𝑗)(3.5)

𝑚bra,n,dcc,𝑖,𝑖∗=

𝑄bra,n,dcc,𝑖,𝑖∗

𝑐pΔ𝜗n

(3.6)

In the fourth step, the pipe diameters for the branch 𝑑in,bra,cc,𝑖,𝑖∗and connections 𝑑in,con,𝑖,𝑗

are determined based on the nominal mass flow rates calculated in the second and third

3.1. Model Design

steps. Please note that the index “cc” represents either SCC or DCC. The pipe diameters

are calculated for each pipe segment and network expansion. The inner pipe diameter can

be designed using either continuous diameters or by selecting available diameters from

the pipe manufacturer’s specifications. For continuous diameter calculations, the inner

pipe diameters are derived to achieve the given nominal pressure loss Δ𝑝nat a given

pipe roughness 𝑟. The resulting Equation is obtained by neglecting terms 1and 3of

Equation (2.20) and rearranging the resulting equation to 𝑑in, which leads to Equations

(3.7) and (3.8). Here, 𝑓∗(𝑑in)corresponds to the pipe friction factor 𝑓∗, which is also a

function of the inner pipe diameter (see Equation (2.21)). Unfortunately, Equations (3.7)

and (3.8) cannot be solved analytically and must be solved iteratively.

𝑑5

in,bra,cc,𝑖,𝑖∗=8𝑚2

bra,n,cc,𝑖,𝑖∗

𝜋2𝜚Δ𝑝n𝑓∗(𝑑in,bra,cc,𝑖,𝑖∗)(3.7)

𝑑5

in,con,𝑖,𝑗 =8𝑚2

con,n,𝑖,𝑗

𝜋2𝜚Δ𝑝n𝑓∗(𝑑in,con,𝑖,𝑗)(3.8)

The inner and outer diameters of the corresponding pipe insulation for the main branch

𝑑out,bra,cc,𝑖,𝑖∗and connection pipes 𝑑out,con,𝑖,𝑗 are linearly interpolated according to the

corresponding inner pipe diameter, the pipe insulation class IC, and the corresponding

pipe specifications of a pipe manufacturer. Tables A.8 – A.10 provides an overview of

selected pipe manufacturer data. If the inner pipe diameters are determined using the

pipe manufacturer’s specifications, the continuous diameter is calculated first, and the next

larger available pipe diameter is selected from the manufacturer’s specifications afterward.

After the completion of step four, all pipe segments 𝑖∗for all possible network ex-

pansions 𝑖were designed for both SCC and DCC. With this, all necessary data is now

available to perform the techno-economic calculations for the linear DHN in step five. This

involves computing the heat losses, i.e., nominal heat loss flow and annual heat loss, of

all pipe segments for all network expansions ( 

𝑄loss,bra,n,cc,𝑖,𝑖∗,

𝑄loss,con,n,𝑖,𝑗,𝑄loss,bra,a,cc,𝑖,𝑖∗,

𝑄loss,con,a,𝑖,𝑗) in accordance with Equations (2.27) and (2.30). Thereafter, the total heat

losses are determined for each network expansion given by Equations (3.9) and (3.10).



𝑄loss,n,cc,𝑖 =𝑖

∑

𝑖∗=1(

𝑄loss,bra,n,cc,𝑖,𝑖∗+𝑛rad

∑

𝑗=1 

𝑄loss,con,n,𝑖∗,𝑗)(3.9)

𝑄loss,a,cc,𝑖 =𝑖

∑

𝑖∗=1(𝑄loss,bra,a,cc,𝑖,𝑖∗+𝑛rad

∑

𝑗=1𝑄loss,con,a,𝑖∗,𝑗)(3.10)

Once the heat losses have been calculated, it is possible to determine the nominal power

and annual produced heat of the heat generator using Equations (3.11) and (3.12).



𝑄hg,n,cc,𝑖 =

𝑄loss,nw,cc,𝑖+𝑖

∑

𝑖∗=1

𝑛rad

∑

𝑗=1 

𝑄con,n,𝑖∗,𝑗 (3.11)

𝑄hg,a,cc,𝑖 =𝑄loss,nw,a,cc,𝑖+𝑖

∑

𝑖∗=1

𝑛rad

∑

𝑗=1𝑄con,a,𝑖∗,𝑗 (3.12)

In the following step, the nominal pressure losses of each pipe segment Δ𝑝loss,bra,a,𝑖,𝑖∗

and Δ𝑝loss,con,n,𝑖,𝑗 are determined for each network expansion by utilizing Equation (2.20).

Here, the term 1is neglected, as the geodetic pressure difference is insignificant when

estimating the pressure loss. The pressure loss of all fixtures is assumed to be a fixed

percentage of the pressure loss of the straight pipe, represented by 𝜉n. As suggested by

3. Detailed Simulation Model

Reference [70], values of 𝜉n=0.1...0.2are reasonable. Therefore, Equation (2.20) can be

expressed as Equation (3.13) for pressure losses of the main branch and as Equation (3.14)

for pressure losses of the connections.

Δ𝑝loss,bra,n,cc,𝑖,𝑖∗=𝜚

2𝑣2𝑓∗Δ𝐿bra,𝑖∗

𝑑in,bra,cc,𝑖,𝑖∗(𝜉n+1) (3.13)

Δ𝑝loss,con,n,𝑖,𝑗 =𝜚

2𝑣2𝑓∗Δ𝐿con,𝑖,𝑗

𝑑in,con,𝑖,𝑗 (𝜉n+1) (3.14)

In the subsequent step, the pressure increase of the pump is determined by adding the

pressure loss of the supply and return pipe along the critical path to the pressure loss of the

substation. The critical path is defined as the path characterized by the largest pressure

loss from the heat generator to a consumer, which is usually the consumer furthest away

from the heat generator. In the context of a linear DHN, the critical path corresponds to

the pressure loss of the entire main branch plus the pressure loss of the last connection pipe

and its substation as described by Equation (3.15). If the number of radial connections is

greater than one, only the consumer with the greatest pressure loss is considered, which

is achieved through the use of the max()operator in Equation (3.15). The factor 2 in the

first and second therm in Equation (3.15) is related to the two-pipe system which consists

of a supply and return pipe.

Δ𝑝pump,n,𝑖 =2 𝑖

∑

𝑖∗=1Δ𝑝loss,bra,𝑖,𝑖∗+2max (Δ𝑝loss,con,𝑖)+Δ𝑝sst,min (3.15)

The nominal electrical power consumption of the pump 𝑃el,pump,n,𝑖 is calculated using

Equation (3.16). Furthermore, the annual power consumption of the pump is estimated

using Equation (3.17), which is discussed in Section 2.4.5 as an approximation of the

annual power demand.

𝑃el,pump,n,cc,𝑖 =

𝑄hg,n,cc,𝑖Δ𝑝pump,n,𝑖

𝜚𝑐pΔ𝜗n𝜁pump

(3.16)

𝑊el,pump,a,cc,𝑖 =𝑃el,pump,n,cc,𝑖𝜏full (3.17)

In large-scale expansions of the linear DHN, it may be necessary to allocate the total

pressure increase among multiple pumping stations to prevent exceeding the maximum

operating pressure (MOP). The number of pumping stations required can be calculated

using Equation (3.18). It is important to note that the brackets ⌈⌉in Equation (3.18)

denote the ceiling function, which returns the smallest integer greater than or equal to the

input. 𝑛pump,𝑖 =⌈Δ𝑝pump,n,𝑖+𝑝R

𝑝max ⌉(3.18)

Once all the required technical quantities have been calculated, the economic quantities

can be determined. The first step in this process is to calculate the investment costs of

the network components. This includes the investment costs of all pumping stations

𝐶inv,pump,𝑖, piping network 𝐶inv,pip,cc,𝑖, and substations 𝐶inv,sst,𝑖. The investment costs of

the piping network are determined by adding up the investment costs of all pipe segments

in the branch and connection according to Equation (3.19).

𝐶inv,pip,cc,𝑖 =𝑖

∑

𝑖∗=1Δ𝐿bra,𝑖∗(𝐼0+𝐼1𝑑in,bra,cc,𝑖,𝑖∗)+ 𝑖

∑

𝑖∗=1

𝑛rad

∑

𝑗=1Δ𝐿con,𝑖∗,𝑗 (𝐼0+𝐼1𝑑in,con,𝑖∗,𝑗)

(3.19)

3.2. Case Studies and Input Parameters
The investment costs of all pumping stations can be calculated using Equation (3.20),
while the investment costs of all substations can be determined using Equation (3.21).
𝐶inv,pump,cc,𝑖 =
𝑄hg,n,cc,𝑖 𝑐inv,pump 𝑛pump,𝑖 (3.20)
𝐶inv,sst,𝑖 =𝑖
∑
𝑖∗=1
𝑛rad
∑
𝑗=1 
𝑄con,𝑖∗,𝑗 𝑐inv,sst (3.21)
In the next step, partial LCOH are calculated for capital, heat loss, pumping, and
O&M. Then, overall LCOH are calculated. Capital costs are calculated for each possible
network expansion using Equations (3.22)–(3.25).
𝑐cap,dst,cc,𝑖 =𝑐cap,pip,cc,𝑖+𝑐cap,pump,𝑖+𝑐cap,sst,𝑖 (3.22)
𝑐cap,pip,cc,𝑖 =𝑎𝐶inv,pip,cc,𝑖
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.23)
𝑐cap,pump,𝑖 =𝑎𝐶inv,pump,𝑖
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.24)
𝑐cap,sst,𝑖 =𝑎𝐶inv,sst,𝑖
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.25)
Pumping LCOH are computed using Equation (3.26), heat loss LCOH are computed for
each network expansion using Equation (3.27), O&M LCOH are computed using Equation
(3.28), and the overall LCOH of heat distribution are calculated using Equation (3.29).
𝑐pump,dst,cc,𝑖 =𝑊el,a,pump,cc,𝑖
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.26)
𝑐loss,dst,cc,𝑖 =𝑄loss,nw,a,cc,𝑖
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.27)
𝑐om,dst,cc,𝑖 =𝑄hg,a,cc,𝑖 𝑐om,var,dst +
𝑄hg,n,cc,𝑖 𝑐om,fix,dst
∑𝑖𝑖∗=1∑𝑛rad
𝑗=1 𝑄con,a,𝑖∗,𝑗 (3.28)
𝑐dst,cc,𝑖 =𝑐cap,dst,cc,𝑖+𝑐pump,dst,cc,𝑖+𝑐loss,dst,cc,𝑖+𝑐om,dst,cc,𝑖 (3.29)
Thus, all the necessary technical and economic quantities have been calculated, and
they can be visualized and exported in the final step six before the algorithm will be
terminated.
3.2 Case Studies and Input Parameters
This section outlines the case studies and their corresponding boundary conditions, which
enable the analysis of distribution costs predicted by the detailed simulation model. The
primary goal of these investigations is to gain an understanding of how the input param-
eters of a linear DHN affect the LCOH. In total, four investigations are conducted using
the detailed model:
1. Validation study
2. Default model analysis
55

3. Detailed Simulation Model

3. OFAT parameter study

4. Monte Carlo parameter study

In the first step, a validation study is conducted to ensure that the predicted LCOH

obtained from the detailed model is consistent with the data reported in the literature. The

purpose of validation is to verify and confirm the correct operation of the model against

the accurate data for assessing the knowledge obtained from the model [97]. To accomplish

this, two databases with their respective advantages and disadvantages are used for the

validation. The model parameters are selected based on the validation databases and are

significantly different from those used in the remaining three investigations. The detailed

information about the selected model parameters is presented in Section 3.3.

For the default model analysis, as well as the OFAT and Monte Carlo parameter

studies, a set of input parameters was selected and used consistently. Table 3.2 presents

an overview of the input parameters that were chosen and grouped into variable, constant,

and binary categories. Each variable input parameter was assigned to a default value, as

well as a minimum and maximum value to allow for sensitivity analysis. For constant

and binary input parameters, only a default value was given. Furthermore, additional

parameters could be derived from the selected input parameters. Each derived input

parameter was assigned a default value, as well as a minimum and maximum value, based

on the range of variable input parameters used to derive it. The selected range of variable

input parameters was designed to cover a diverse range of real-world DHN applications.

The default model analysis consists of two parts. Firstly, only the default values

presented in Table 3.2 were used as input parameters, and consumers were assumed to

be equally distributed and sized. This analysis was conducted to investigate the primary

characteristics of the LCOH concerning network expansion and will be discussed in detail

in Section 3.4.1. Secondly, the default input parameters were again used, but the effect of

non-equally distributed and sized consumers was analyzed, as presented in Section 3.4.2.

In addition to the default model analysis, an OFAT sensitivity study was conducted in

Section 3.5 to examine the impact of a single input parameter on the LCOH distribution

and its sensitivity. The OFAT method was employed, where a single parameter is varied

while keeping all other input parameters at their default values. This method allows for an

isolated investigation of individual parameters influence. However, the drawback is that

the sensitivity can only be analyzed based on a single default scenario, and the interaction

among different parameters cannot be considered.

To investigate the interaction of input parameters and their combined effect on the

LCOH, a Monte Carlo study was conducted in Section 3.6. This involved a large number

of simulations where variable input parameters were set randomly within the specified

limits, as presented in Table 3.2. By doing so, the study aimed to examine how different

input parameter combinations influence the LCOH, and identify the favorable conditions

that must be met to achieve the desired outcome. The outcomes of the Monte Carlo

analysis were utilized to develop regression models that can forecast the LCOH based on

the provided network design parameters.

3.2. Case Studies and Input Parameters

Table 3.2: List of default, minimum and maximum model input parameters.

Parameter Unit Default Minimum Maximum

Variable:



𝑄con,n,𝑖,𝑗 (OFAT) [kW] 43 21.5 86



𝑄con,n,𝑖,𝑗 (Monte Carlo) [kW] [-] 5.375 86

𝜏full,con,𝑖,𝑗 [h/a] 2000 1000 4000

𝜗s,n[°C] 80 40 120

𝜗amb [°C] 10 5 15

Δ𝑝n[Pa/m] 250 50 450

𝑟[mm] 0.01 0.001 0.1

𝑝R[bar] 3.0 1.5 6.0

𝑝max [bar] 16 6 25

Δ𝑝min,sst [bar] 0.8 0.4 1.6

𝜉n[-] 0.2 0.0 0.4

𝜂pump [-] 0.7 0.5 0.9

𝜏op [h/a] 8760 4380 8760

𝜆ins [W/m/K] 0.03 0.01 0.06

𝜆soil [W/m/K] 3.0 0.5 6.0

IC [-] 2 1 3

𝐿dep [m] 0.8 0.4 1.6

Δ𝐿con,𝑖,𝑗 [m] 15 5 45

𝐼1[€/m2] 6000 3000 9000

IR [%/a] 4 2 8

𝜏inv [a] 30 15 45

𝑐∗el [€/kWh] 20.0 10.0 40.0

𝑐th [€/kWh] 8.0 4.0 16.0

𝑐inv,pump [€/kW] 90 45 180

𝑐inv,sst [€/kW] 100 50 200

Constant:

𝑛rad [-] 2

Δ𝐿bra,𝑖 [m] 56 --

𝐿bra,tot [km] 10 - -

𝐼0[€/m] 180 - -

𝑐om,var,dst [ct/kWh] 0.15 - -

𝑐om,fix,dst [ct/kWh] 0.0 - -

Binary:

Continuous pipe diameters [-] True

Use SF [-] True

Derived:

Δ𝜗n[K] 27.5 8.4 56.6

𝑞lin (OFAT) [MWh/m/a] 2.0 1.0 4.0

𝑞lin (Monte Carlo) [MWh/m/a] - 0.04 5.21

𝑞lin (OFAT) [kW/m] 1.0 0.5 2.0

𝑞lin (Monte Carlo) [kW/m] - 0.02 2.61

𝑎[%/a] 5.78 4.46 8.99

3. Detailed Simulation Model

3.3 Model Validation

Model validation is a crucial process to compare the performance of a model with the real

system it is meant to represent. The aim of this process is to confirm and demonstrate the

model’s ability to accurately simulate the system’s behavior by comparing the simulation

results with correctly classified data. This allows the model’s reliability to be evaluated

and provides valuable insights into the underlying processes [97]. In the context of the

model presented in this chapter, the detailed simulation model’s predictions are to be

compared to the validation data.

3.3.1 The Data Base

The research for appropriate validation data proved to be a challenging task as it requires

both the corresponding input parameters and the target values. However, two suitable

sources were identified in the literature, each with its advantages and disadvantages re-

garding the quality of validation.

The first database is derived from a published model that calculates the LCOH for

capital, pumping, and heat loss costs, which was developed by Nussbaumer and Thalmann

[95, 98]. Using this model for validation has the advantage of using the same set of input

parameters to compare the corresponding target values, enabling both quantitative and

qualitative model validation. However, the disadvantage is that the validation model, even

though it is based on validated methods, does not represent an overall real case scenario.

To overcome this limitation, a second database was utilized that corresponds to real-

life data of constructed DHNs. The disadvantage of this database is that only limited

information on underlying input parameters is available, making it only possible to perform

a qualitative validation. The validation data used here are the network investment costs

and the percentage annual heat losses of a DHN, which are a function of the LHD shown

in Figures 1.2a and 1.2b.

The validation quality is estimated to be suﬀicient but rather coarse when using only

one of the described databases due to the limitations of each. The trustworthiness of the

model is increased when the validation against both databases shows a similar outcome.

Conversely, the trustworthiness of the model is expected to decrease when the validation

against both databases shows different outcomes.

3.3.2 Validation Against Published Model

The model used for validation was initially published in Reference [95] and is available for

download from Reference [98]. The published model was originally developed to conduct a

sensitivity analysis on distribution costs for district heating networks. As such, it assumes

a single consumer supplied by a two-level DHN. The model calculates the capital, pumping,

and heat loss annual costs, as well as the corresponding LCOH, based on a set of input

parameters and for a range of available KMR pipe diameters, DN 20...250

Table 3.3 provides an overview of the matching and mismatching input parameters

of the detailed and published models. The matching input parameters indicate that the

basic functionality of the detailed model can be tested against the published model. The

mismatching input parameters in the published model, namely 𝜗r,n,𝜁el,𝜁hyd,𝑐∗f, and 𝜉𝑖,

can be derived from the input data of the detailed model. For instance, the nominal return

temperature 𝜗r,ncan be derived from the nominal supply temperature 𝜗s,nand the nominal

temperature difference Δ𝜗n. However, the investment costs of pipes are treated slightly

differently in both models. The published model uses defined specific pipe investment

costs 𝑐inv,pip for each pipe diameter, while the detailed model uses a linear approximation

3.3. Model Validation

function to estimate the specific piping costs using the coeﬀicients 𝐼0and 𝐼1. Nonetheless,

when the coeﬀicients of 𝐼0and 𝐼1are calculated according to the given investment costs

used by the published model, the differences are expected to be minor.

Moreover, the published model does not take into account certain components of the

detailed model, including investment costs and pressure losses of the connection pipes,

operation and maintenance costs, investment costs of the substations, simultaneity fac-

tor, investment costs of pumping stations, an operation duration other than a full year

operation, and a thermal resistance of two pipes interacting with each other (see term

3of Equation (2.27)). Additionally, the published model uses constant water properties

at a reference temperature of 𝜗ref =60°C, while the detailed model estimates the refer-

ence temperature according to the nominal temperatures and the corresponding resting

pressure. Furthermore, the correlation used to estimate the pipe friction factor 𝑓∗differs

in both models. Nonetheless, the limitations of the published model in comparison to

the detailed model are minor, which allows a validation of the major components of the

detailed model.

Table 3.3: Matching and mismatching input parameters of the detailed and the published

model.

Matching 

𝑄con,n,𝑖,𝑗,𝜏full,con,𝑖,𝑗,𝜗s,n,Δ𝐿bra,𝑖, IR, 𝜏inv,𝑐∗el,𝑟,Δ𝑝sst,min,

𝜗amb,𝜆ins,𝜆soil,𝐿dep

Mismatching Detailed model Published model

Δ𝜗n,Δ𝐿con, SF, 𝜁pump,Δ𝑝n,

𝑝R,𝜉n,𝑝max,𝜏op,𝐼0,𝐼1,

𝑐th,𝑐om,var,dst,𝑐om,fix,dst,𝑐th,

𝑐inv,pump,𝑐inv,sst

𝜗r,n,𝜁el,𝜁th,𝜁hyd,𝑐∗f,𝜉𝑖,

𝑐inv,pip

To validate the detailed model against the published model, a use-case was defined and

adapted to both models. The use-case represents a scenario of a linear DHN, where connec-

tion pipes and investment costs for pumping stations and substations are not considered.

The total branch length of the network is assumed to be 𝐿bra,tot =10km, which is divided

into 100 pipe segments. The LHD is assumed to be constant at 𝑞lin =2.0MWh/m/a.

The nominal supply and return temperatures are selected in such a way that the reference

temperature is 𝜗ref =60°C, which leads to similar water properties for both models. It

is assumed that the pipes are buried within a street. All the remaining input parameters

are listed in Table 3.4.

Table 3.4: Input parameters of the use-case to validate the detailed model against the

published model.

Parameter 

𝑄con,n,𝑖,𝑗 𝜏full,con,𝑖,𝑗 Δ𝐿bra,𝑖 𝜗s,nΔ𝜗nΔ𝑝n𝑟 𝜆ins 𝜆soil

Unit [kW] [h/a] [m] [°C] [K] [Pa/m] [mm] [W/m/K] [W/m/K]

Value 100 2000 100 70 20 200 0.01 0.026 1.2

Parameter Δ𝑝min,sst 𝜁pump SF 𝜗amb 𝐿dep IR 𝜏inv 𝑐∗el 𝑐th

Unit [bar] [−] [−] [°C] [m] [%/a] [a] [ct/kWh] [ct/kWh]

Value 1 0.75 1 10 0.6 4.0 30 20.0 8.0

Parameter 𝐼0𝐼1𝜁el 𝜁hyd 𝜁th 𝑐∗f𝜉𝑖Δ𝐿con,𝑖,𝑗

Unit [€/m] [€/m2] [−] [−] [−] [ct/kWh] [−] [m]

Value 219.4 6246.7 1.0 0.75 1.0 8.0 0.0 0.0

The validation detailed model against the published model are presented in Figure 3.3.

3. Detailed Simulation Model

The figure displays the variation in the LCOH components, including capital, heat loss,

pumping, and total costs, as a function of the branch length 𝐿bra. It is important to note

that the branch length in Figure 3.3 corresponds to a specific network expansion length.

Table 3.5 shows the corresponding root mean squared error (RMSE) and normalized root

mean squared error (NRMSE) values, where RMSE refers to the total and NRMSE to

the relative error. The NRMSE value is the RMSE normalized by the mean value of

the results obtained from the published model. Overall, the two models demonstrate

good agreement. The relative error for the total costs is NRMSE =3.7%. Upon closer

inspection, it is observed that deviations occur for the capital and pumping costs. These

deviations can be attributed to two factors. Firstly, the specific pipe investment cost

calculations differ slightly between the two models. Secondly, different correlations were

used to estimate the pipe friction factor, which leads to variations in pressure losses and

subsequently slightly different branch lengths at which the next possible pipe diameter

is selected. Figure 3.4 displays a detailed plot of the inner pipe diameter and friction

factor. The figure shows that the detailed model estimates smaller friction factors than

the published model, resulting in an increase in pipe diameters at larger branch lengths.

The different specific pipe investment cost calculations used in both models are shown

in Figure 2.23. The linear regression function in the detailed model corresponds to the

published model data point. This variation in approach leads to slightly different piping

capital costs. However, the relative error of the capital costs is NRMSE =5.9%, which

is still within an acceptable range. The relative error of the heat loss LCOH is only

NRMSE =1.32%, which is very low. Conversely, the pumping LCOH demonstrates the

largest relative error of NRMSE =19.13%. This error is associated with the small values

of the pumping costs. The absolute error of the pumping costs is RMSE =0.08ct/kWh,

which is relatively small compared to the total absolute error.

0 2 4 6 8 10

Branch length Lbra [km]

LCOH cdst [ct/kWh]

Total

(detailed)

Total

(published)

Capital

(detailed)

Capital

(published)

Heat loss

(detailed)

Heat loss

(published)

Pumping

(detailed)

Pumping

(published)

Figure 3.3: Distribution of the LCOH related to the network expansion of the detailed

and the published model according to the validation use-case.

In summary, a good correspondence between the detailed model and the published

model can be observed in general, particularly in terms of the capital, heat loss, and

pumping LCOHs. The principle progression of the LCOH in correspondence to the network

expansion also matches well between the two models. The differences that occur are minor,

and can be explained and comprehended.

Table 3.5: Error of the cost prediction using the detailed model compared to results

obtained from the published model.

Evaluation Parameter Unit Total Capital Heat Loss Pumping

RMSE [ct/kWh] 0.2030 0.2397 0.0128 0.0758

NRMSE [%] 3.70 5.86 1.32 19.13

3.3. Model Validation

0 2 4 6 8 10

Length Lbra [km]

0.012

0.014

0.016

0.018

Pipe friction factor f∗[-]

Detailed model

Published model

100

150

200

250

Inner diameter din,bra [mm]

Figure 3.4: Pipe friction factor shown in black (left y-axis) and inner pipe diameter shown

in gray (right y-axis) related to the network expansion of the detailed and the published

model according to the validation the use-case.

3.3.3 Validation Against Real Case Data

The use of real case data represents an ideal validation method as it allows direct compar-

ison of target values to data from actual cases. However, the challenge lies in identifying

suitable data sets that provide the necessary input parameters and target values for vali-

dation. Despite efforts to find a database that provides all the required information, none

were identified. Nonetheless, by combining multiple data sources and making appropriate

assumptions, a qualitative validation could be performed. Two data sets were identified

that can be used to validate the investment costs and annual heat losses. The first data set

provides information on specific investment costs in relation to the LHD and the piping

system used. The second data set provides information on the annual heat losses with

respect to the LHD. Consequently, both data sets can be used to validate the estimation

of specific investment costs and the heat losses.

No suitable data sets were found to allow for a serious qualitative validation of pumping

costs against real data. Nevertheless, quantitative validation of pumping costs against a

published model suggests a good match of pumping costs.

Invest Costs

The validation of the investment cost prediction involved defining a scenario to compare

specific investment costs estimated by the detailed model to actual real case data. Initially,

an investigation of the validation data was conducted. The validation data consisted of

two sub-datasets that provide information on specific investment costs of heat distribution

relative to the corresponding LHD. The first subset is derived from Reference [53] and

presents information on specific investment costs of the heat distribution. This includes

costs for the piping system, trench costs, consumer connection, pumping stations, and

consumer substation. The specific investment costs are presented in €/MWh/a and are

related to the annual heat sold to the consumer. In total, three correlations are given that

correspond to three different piping systems and certain construction conditions. The three

correlations correspond to KMR pipes at unfavorable conditions, KMR pipes at favorable

conditions, and very favorable conditions achievable by using PMR pipes. However, there

is no explicit explanation provided for what exactly is meant by favorable and unfavorable

conditions. Here, it was assumed that favorable laying situations of KMR pipes refer to

laying in open fields, while unfavorable laying situations of KMR pipes refer to a laying

within streets. These correlations were derived from realized district heating systems in

Austria and Switzerland constructed between 2009 and 2018.

The second subset was obtained from a case study conducted in Switzerland, as re-

3. Detailed Simulation Model

ported in Reference [30]. This case study analyzed 52 existing district heating systems

and provides specific investment costs for the heat distribution with respect to the corre-

sponding LHD. Specifically, the subset includes information on specific investment costs

for 36 out of 52 cases, with 22 cases including costs for consumer substations and 14 cases

excluding costs for consumer substations.

To validate the estimated network investment costs of the detailed model, certain input

parameters shown in Table 3.6 were required. Among these, the LHD value could be di-

rectly obtained from the validation dataset, which ranged from 𝑞lin =0.5...6.0MWh/m/a.

The remaining input parameters were based on assumptions. The linear pipe investment

costs coeﬀicients 𝐼0and 𝐼1of KMR pipes were used for open field and street laying,

respectively, based on Figure 2.23. Both laying situations were considered to cover favor-

able and unfavorable piping situations. The average branch length was assumed to be

Δ𝐿bra,𝑖 =59m, which is the average value for DHNs in Germany [99]. However, there

was no reliable information found in the literature on the consumer connection length, and

thus, it was estimated to be Δ𝐿con,𝑖,𝑗 =15m. The full load hours of the consumer were esti-

mated based on an average value of DHS valid for Germany, which was 𝜏full,con =2155h/a

[99]. The nominal consumer power 

𝑄con,n,𝑖 was derived from the received data using Equa-

tion (3.30), resulting in values ranging from 

𝑄con,n,𝑖 =14...164kW. The nominal supply

temperature was determined to be within the range of 𝜗s,n=70...110°C, which was ob-

tained from the same dataset used to obtain the specific investment costs. The given data

of the nominal supply temperature corresponds to the data shown in Figure 2.21, which

was used to determine the correlation between the nominal supply temperature and the

nominal temperature difference. Among the cases analyzed, 40 out of 44 had nominal

supply temperatures below 𝜗s,n=110°C [30].



𝑄con,n,𝑖 =𝑞lin Δ𝐿bra,𝑖

𝜏full,con,𝑖,𝑗 (3.30)

The specific investment costs of consumer substations 𝑐inv,sst were determined based

on the consumer nominal capacity, as per Equation (2.5), and coeﬀicients from Table 2.4

for the average case. The specific investment costs of the pumping station were taken as

𝑐inv,pump =90€/kW, which is an average value suggested by the Danish Energy Agency

[10]. The pipe roughness was assumed to be 𝑟=0.01mm [70]. Moreover, the simultane-

ity factor was estimated with respect to the number of consumers, using Equation (2.7).

The DCC pipe configuration was assumed, as it represents a more realistic scenario. In-

formation on the total trench length of the validation cases under consideration was not

provided, but both datasets corresponded to mainly wood-fired district heating systems

with medium-sized networks having a range of distribution lengths of 𝐿tot =0.5...5.0km

[100, 30]. The nominal design pressure loss and nominal design temperature difference

were both assumed to be within typical ranges, as they significantly influence pipe design

and are dependent on the individual case. The nominal pressure loss was assumed to be

Δ𝑝n=150...300Pa/m, based on typical design values [53, 70, 52]. The nominal design

temperature difference was assumed to be Δ𝜗n=20...40K, which is a typical range of

real-case data [30].

Table 3.6: Required input parameter needed for a validation against real-life data.

Investment costs Heat losses



𝑄con,𝑖,𝑗,𝜏full,con,𝑖,𝑗 𝜗s,n,Δ𝐿bra,𝑖,Δ𝐿con,𝑖,

𝑞lin,Δ𝑝n,𝑟,𝐼0,𝐼1,𝑐inv,pump,𝑐inv,sst, SF 

𝑄con,𝑖,𝑗,𝜏full,con,𝑖,𝑗 𝜗s,n,Δ𝐿bra,𝑖,Δ𝐿con,𝑖,

𝑞lin, SF, IC, 𝜏op,𝜆ins

3.3. Model Validation

The validation results are presented in Figure 3.5. The validation data is divided

into two subsets: real cases and KMR data, which are shown as data points and lines,

respectively. The detailed model’s predictions are represented by two filled areas that

correspond to two branch lengths: 𝐿bra,tot =0.5km and 𝐿bra,tot =5.0km, at which the

detailed model was evaluated. The filled areas represent the resulting target values, which

is a cloud of data due to the consideration of ranges of input parameters 𝐼0,𝐼1,Δ𝑝n, and

Δ𝜗n. This evaluation at different locations was necessary to cover the considered length

range expected from the validation data.

1 2 3 4 5 6

Linear heat density [MWh/m/a]

500

1000

1500

2000

Specific invest costs [€/(MWh/a)]

Area of detailed model results (L= 5.0 km)

Area of detailed model results (L= 0.5 km)

KMR unfavorable conditions

KMR favorable conditions

Real cases incl. substation

Real cases excl. substation

Regression real cases (R2= 0.18439)

Figure 3.5: Specific investment costs for several values of the LHD obtained from the

detailed model compared to real case data. Real case data was derived from Reference

[30].

The results obtained from the detailed model show a similar trend compared to the

validation data (KMR and real cases data). This trend is characterized by lower specific

investment costs at higher LHDs and vice versa. The results obtained from the detailed

model cover the suggested lower bandwidth of the KMR pipes validation data very well,

except for LHDs below 1.0MWh/m/a, where the lower bandwidth suggested by the KMR

pipe data is not reached by the detailed model. The upper limit suggested by the KMR

pipe data is exceeded along the entire observed range of the LHD, which indicates an

overestimation of the specific investment costs by the detailed model when compared to

the KMR data. However, the validation data of the real cases indicate a much wider

bandwidth as expected by the KMR data, indicating a much greater variability of the

specific investment costs in general. Therefore, the overestimation for lower values of the

LHD by the detailed model compared to the KMR data should not be overestimated.

When the detailed model is compared to the data from real cases, the results are

within the cloud of real case data. A regression curve of the data from real cases is fitting

the outcome of the detailed model well. Therefore, the investment cost estimation of the

detailed model is considered valid when validated against the considered validation data.

Heat Losses

The validation of the detailed model against actual data was performed in a similar manner

as the validation of the network investment costs against real data. To achieve this,

a dataset was selected that enables a qualitative validation. This dataset contains the

percentage annual heat loss as a function of the LHD for 330 DHNs. The necessary input

parameters to validate the detailed model are listed in Table 3.6. No further information

besides the LHD could be obtained from the validation dataset. Therefore, typical values

have been selected to provide suﬀicient input parameters for the detailed model. Some

input parameters are expected to have a strong influence on heat losses, while others are

3. Detailed Simulation Model

Values of detailed model up to 91.2% for L = 10.0 km

and values up to 66.1% for L = 0.5 km.

Figure 3.6: Percentage annual heat losses with respect to the LHD obtained from the

detailed model and compared to real case data. Real case data was derived from Reference

[30].

expected to have a weak influence. The parameters assumed to have a strong effect on

heat losses were varied using typical ranges. The parameters assumed to have a weak

effect on heat losses were set according to average values obtained from the literature.

The model input parameters that have a strong influence on the percentage heat losses

are the operational hours 𝜏op, the heat conductivity coeﬀicient of the insulation 𝜆ins, the

insulation class IC, the nominal supply temperature 𝜗s,nand the nominal temperature dif-

ference Δ𝜗n. These values were varied according to typical values from DHNs that are cur-

rently in operation. The operational hours were specified in a range of 𝜏op =182.5...365d/a

to cover systems with seasonal and year-round operation. The heat conductivity coeﬀi-

cient of the insulation was varied in a range of 𝜆ins =0.02...0.04W/m/K, which covers

a range of available insulation materials, including material aging effects [101, 102]. The

insulation class was varied in a range of IC =1...3to cover all available insulation thick-

nesses offered by pipe manufacturers. The supply temperature was varied in a range of

𝜗s,n=70...110°C to cover typical supply temperatures of third-generation DHNs that are

in operation today [30, 6]. The temperature spread was varied according to the nominal

supply temperature in a range of Δ𝜗n=20...52K [30].

The total length of the network branches was set to 𝐿bra,tot =10.0km, in order to

simulate a broad range of network lengths that are typical in practice. The ambient

temperature was chosen to correspond to the average ambient temperature of Germany

in 2020, namely 𝜗amb =10.6°C, which is consistent with the year the validation data was

obtained [103, 99]. The remaining input parameters were set to the same values as those

used to validate the investment costs.

The results of the study are presented in Figure 3.6. The validation data, representing

real cases, are plotted as a scatter plot, while the results obtained from the detailed models

are shown as filled areas. The filled areas correspond to network lengths of 𝐿bra,tot =0.5km

and 𝐿bra,tot =10.0km. The results indicate that higher values of the LHD are associated

with lower annual heat losses in the validation data. The validation data also exhibit

significant variance, with greater variance observed at lower LHDs. The results obtained

from the detailed model align well with this range of variance. Specifically, the upper and

lower limits of the validation data are well captured by the model. However, a few outliers

are observed between 𝑞lin =3...5MWh/m/a, which might be due to non-ideal design and

operation conditions. Furthermore, the detailed model reveals annual heat losses of up to

91.1%at 𝑞lin =0.5MWh/m/a and 𝐿bra,tot =10.0km. However, this value is significantly

higher than the maximum observed in real cases at low LHDs (𝑞lin <1.0MWh/m/a). The

3.4. Default Model Investigations

high heat losses are unlikely to occur in real cases because unfavorable design conditions

are typically avoided by engineers during design phases. On the contrary, the lower limit

of the detailed model aligns well with the results of real cases. Engineers typically prefer

design conditions that lead to low annual heat losses, and therefore, real cases typically

align with the lower limit of the detailed model.

The validation of the detailed model against real cases demonstrates a satisfactory

agreement in general. Both the data points and the corresponding regression line of the

real cases are adequately covered by the results of the detailed model. Therefore, the

qualitative validation of the detailed model against real data can be assumed successful.

3.4 Default Model Investigations

This section presents and discusses investigations carried out using the default model

parameters. The distribution costs and their shares are investigated in detail as a function

of the network expansion of the linear DHN in Section 3.4.1. This investigation uses a

constant LHD, but in real DH applications, variable distributions of the LHD may be

possible. Hence, a second investigation is presented in Section 3.4.2, where the effects of

non-uniform LHDs along the linear network path are analyzed using the default model

parameters.

3.4.1 General Investigations

Figure 3.7 shows the LCOH distribution along the linear network path as obtained from

the default model. The figure presents the capital, heat loss, pumping and O&M LCOH for

SCC and DCC. In addition, the capital costs estimated using Persson’s method introduced

in Section 1.2.3 are also shown, which was computed using Equations (1.10) and (2.42)

for the same input parameter as the default model. The relationship between the specific

costs and the network length, 𝑐dst =𝑓(𝐿bra), is referred to as the characteristic function

in this thesis. It is worth noting that the branch length 𝐿bra corresponds to a certain

network expansion until that particular point.

The analysis shows that the characteristic functions of the total LCOH exhibit a mono-

tonically increasing trend with respect to branch length for both SCC and DCC. However,

the total LCOH values are higher for SCC along the entire path compared to DCC. The to-

tal LCOH values range from 2.4...7.1ct/kWh for SCC and from 2.4...6.4ct/kWh for DCC.

It is observed that the LCOH more than double along the network path for both config-

urations, which indicates that the network length plays a significant role in determining

the LCOH. The relative error between the trends of the total LCOH for SCC and DCC is

0 2 4 6 8 10

Branch length Lbra [km]

LCOH [ct/kWh]

Total (SCC)

Total (DCC)

Capital (SCC)

Capital (DCC)

Heat loss (SCC)

Heat loss (DCC)

Pumping (SCC)

Pumping (DCC)

OM (SCC+DCC)

Capital (Persson)

Figure 3.7: Distribution of the LCOH related to the network expansion derived from the

default parameter set of the detailed model.

3. Detailed Simulation Model

presented in Table 3.7, and the relative error between the SCC and DCC distribution is

NRMSE =10.4%.

Table 3.7: Relative and absolute error between the DCC and SCC LCOH distribution.

Data obtained from the detailed model using the default parameter set.

Evaluation parameter Unit Total Capital Heat Loss Pumping O&M

RMSE [ct/kWh] 0.5236 0.4058 0.1163 0.0 0.0

NRMSE [%] 10.37 13.00 11.98 0.0 0.0

Upon examination of the different components of the total LCOH, it is apparent that

the capital LCOH makes up the majority of the cost along the entire network path for

both SCC and DCC. The pumping, heat loss, and O&M LCOH constitute a significantly

smaller portion of the total cost.

The distribution of the capital LCOH scales with network expansion because a larger

network expansion results in larger average pipe diameters. Assuming a constant heat

density, this leads to a consistent amount of heat that can be sold per network expansion.

However, since a greater network expansion also leads to a higher nominal thermal power,

the pipe diameters need to be enlarged to accommodate the increased thermal load. As

larger pipe diameters are associated with higher costs, a network expansion results in larger

specific heat distribution capital costs. The shape of the caracteristic function shown in

Figure 3.7 demonstrate that pipe capital costs and network expansion are related through

a root function.

Moreover, longer networks exhibit greater pressure losses, necessitating additional

pumping stations because the network’s maximum pressure is limited to the maximum

operational pressure. This leads to increased capital LCOH for longer networks. This

effect can be recognized in the distribution of the capital LCOH costs shown in Figure

3.7. Here, the distribution is marked by various discontinuities for both configurations,

which is related to increased pressure losses for longer networks. As the maximum pres-

sure of the pipes is limited by the maximum operational pressure, the required differential

pressure needs to be distributed among several pumping stations if the network expands.

Not considering that would lead to exceeding the maximum operational pressure, which

is not permitted. Each pumping station requires additional capital costs, leading to a

discontinuous increase in capital costs. It is important to note that the discontinuities of

the capital costs shown in Figure 3.7 are not related to the discontinuous availability of

pipe diameters, since the pipe diameters of this model were described as continuous. Fur-

thermore, it is observed that the capital LCOH is lower for DCC than for SCC, achieving

aNRMSE of 13.0%compared to each other (see Table 3.7). This is because the average

pipe diameter is lower for DCC, which leads to lower pipe investment costs.

The comparison between the capital LCOH according to Persson’s method and the

results of the detailed model indicate acceptable consistency at lower network lengths

𝐿bra <2km for both SCC and DCC. However, the effect of increasing capital LCOH due

to network expansions leads, particularly for longer network expansions, to unacceptable

differences between the capital costs according to the detailed model and Persson’s method.

The LCOH of heat losses exhibits a slightly increasing trend at small trench lengths,

where 𝐿bra <0.5km, and an almost constant trend at higher trench lengths. The relative

error between SCC and DCC is comparable to the relative error of the total LCOH, with

a value of NRMSE =12.0%. Upon comparing the heat loss LCOH to the pumping and

capital LCOH, an approximately constant trend can be observed. This is because the

absolute heat losses increase with expanding the network, but the heat demand of all

3.4. Default Model Investigations

consumers also increases simultaneously. As a result, the relative heat losses and costs are

only dependent on the ratio of the outer and inner insulation diameter of the pipes, as

expressed by Equation (2.50). This ratio changes more significantly with smaller diameters,

while it changes only slightly for larger pipe diameters. The heat loss LCOH of the DCC

exhibits a slightly lower trend than that of the SCC. This difference is attributed to the

smaller average pipe diameters of the DCC, which results in a smaller area available for

heat transfer at smaller pipe diameters and hence slightly smaller heat losses.

The pumping LCOH in Figure 3.7 exhibits a linear increase with respect to the total

network length for both configurations. This increase is primarily due to the pressure loss

in the pipes, which increases linearly with network expansion. Therefore, the pumping

LCOH may be insignificant for small network lengths but should be considered for larger

network expansions. The trend of pumping LCOH is the same for both configurations, as

the piping system is designed according to the same nominal pressure loss.

The O&M LCOH distribution is constant and similar for both SCC and DCC. This

observation corresponds to Equation (2.52) and the specific variable O&M costs 𝑐om,var,dst.

The share of O&M LCOH in total costs is negligible (< 2%), which is why they are

excluded from the following investigations.

In summary, the investigations of the default model have shown that network expan-

sion has a significant impact on the specific heat distribution costs. These costs increase

with network expansions due to the increased specific capital and pumping costs. Spe-

cific heat loss and O&M costs do not change significantly concerning network expansions.

Furthermore, the selected configuration has a significant effect on the specific distribution

costs. Assuming a SCC leads to an overestimation of capital and heat loss costs compared

to the DCC. Pumping and O&M LCOH are not affected by the selected configuration.

Considering the total specific costs of the given example, a relative error of about ten

percent between the SCC and DCC was observed.

3.4.2 Influence of Non-Equally Distributed and Non-Equally Sized

Consumers

The investigations conducted in the previous Section 3.4.1 investigated the LCOH at a

LHD of 𝑞lin =2.0MWh/m/a assuming that all consumers and network segments were of

equal size. However, in real-world DHN applications, the nominal power of consumers

and the length of network segments are likely to vary, which can still lead to a LHD of

𝑞lin =2.0MWh/m/a of the entire network.

Therefore, the influence of variable-sized network segments and differently sized con-

sumers an its affects on the LCOH is investigated n this section. For all investigations

carried out, the resulting LHD leads to the same value of 𝑞lin =2.0MWh/m/a. The inves-

tigations are conducted only for a fully expanded network and assuming a DCC. The SCC

is not considered here because variations in the nominal consumer power and the length

of pipe segments do not impact the distribution costs, as long as the total network length

and total consumer load remain constant.

Regarding the nominal consumer power, two possible variations along the linear direc-

tion of the network are possible that would lead to the same LHD of 𝑞lin =2.0MWh/m/a

of the entire network. In the first variation, the nominal consumer power is related to a

scenario in which the distribution is not constant but varies linearly with respect to the

network length. Here, a linearly rising or falling distribution can be considered as shown

in Figure 3.8a. Such distributions of the nominal consumer power can occur if the heat

density changes continuously along the linear direction of the network. A typical example

of such a scenario is related to a case in which an heat generator is located in the city

center and the distribution network is extended to areas that are less densely populated

3. Detailed Simulation Model

on the outskirts of the city. In this case, the consumer connection load is expected to de-

crease at outer areas. However, scenarios with increasing nominal consumer powers along

the linear network path are also possible. This can occur if the heat generator is located

on the outskirts of a city, and the distribution network supplies more densely populated

areas located in the city center.

Mean

Rising

Falling

(a) Linear variation of the

nominal consumer power.

Upper limit

Mean

Lower limit

Bandwidth

(b) Randomly distributed

nominal consumer power.

Mean

ment lengths.

Figure 3.8: Distribution of the nominal consumer power related to the network expansion.

To investigate the influence of the distribution of nominal consumer powers, an equally

sized network1and a linear variation of the nominal consumer power was considered. Here,

four different variations were investigated, of which two represent rising and two represent

falling distributions of the nominal consumer power. The corresponding values are shown

in Figure 3.9. The investigation of the distribution of the nominal consumer power is

named “linear variation of the nominal consumer power”.

0 2 4 6 8 10

Branch length Lbra [km]

Nominal consumer

power ˙

Qcon,n[kW]

Rising 1

Rising 2

Falling 2

Falling 1

Default

Figure 3.9: Variation of the nominal consumer power with respect to the network expan-

sion of the scenario linear changing nominal consumer power.

In the second scenario investigated, the nominal consumer power of each consumer

was randomly varied around a certain mean value along the linear network path, which

is shown in Figure 3.8b. The bandwidth of allowed variations of the nominal consumer

power can be described by the degree of variability DOV. This parameter defines how

much a certain variable can vary in relation to its mean value, and it is calculated according

to Equation (3.31). Four different values of DOV ranging from DOV =0.25...1.0were

considered in the following scenario, which was named “randomly distributed nominal

consumer power”.

DOV =𝑋max −𝑋

𝑋=∣𝑋min −𝑋

𝑋∣(3.31)

Additionally, a scenario was investigated in which the length of each branch segment

was randomly varied. An example of this distribution is shown in Figure 3.8c. The

1It was assumed that the length of pipe segments in both the network branch and the consumer

connection were uniform.

3.4. Default Model Investigations

nominal consumer power was kept at the mean value according to the default model,

while the variability of the segment lengths was varied using DOV values ranging from

DOV =0.25...1.0. Furthermore, the total network length and the total number of segments

were kept constant at the values of the default model. This scenario is named “randomly

distributed segment lengths”.

The default model parameters given in Table 3.2 were adopted as the remaining input

parameters for all simulated scenarios. The investigation of the total LCOH of the DCC

was carried out for each scenario, considering a fully extended network of 𝐿bra =10km.

The obtained results were then compared with those obtained from the default model. It

is important to note that all investigated scenarios are characterized by the same LHD

of 𝑞lin =2.0MWh/m/a, which is similar to the default model. As a result, the relative

change in the total LCOH for a fully expanded network of each case was compared to the

results obtained from the default model. The results of all three introduced scenarios are

shown in Figure 3.10.

−10 −5 0 5

X[%]

Rising 1

Rising 2

Falling 2

Falling 1

Variation

(a) Linear variation of the nom-

inal consumer power.

012

X[%]

0.25

0.5

0.75

DOV

(b) Randomly distributed nom-

inal consumer power.

0.0 0.1 0.2

X[%]

0.25

0.5

0.75

DOV

ment lengths.

Figure 3.10: Results of the investigations according to non-equally distributed and non-

equally sized consumers. Relative difference between total LCOH for a fully extended

network related to the detailed model using the default parameter set. 𝑋 = (𝑐tot −

𝑐tot,default)/𝑐tot,default.

Based on the results of the scenarios with a linear variation of the nominal consumer

power shown in Figure 3.10a, the relative difference compared to the default model varies in

the range of (𝑐tot −𝑐tot,default)/𝑐tot,default =−9.8...3.4%. This implies that the distribution

of the nominal consumer power, or the LHD respectively, impacts the total costs of the

default model. A rising distribution leads to higher total LCOH, while a falling distribution

results in lower total LCOH.

The difference in costs can be attributed to the variation in pipe diameters along

the linear network path. Figure 3.11 presents the inner pipe diameter at full network

expansion for all four scenarios and the default model. Here, it is observed that the

inner pipe diameter decreases with increasing network length, which is associated with

the volume flow extracted by consumers along the linear distribution path. The degree to

which the internal pipe diameter is reduced depends on the size of the consumers along

the path. In the scenario where the nominal consumer power increases, the volume flow

extracted by the consumers is low at shorter network lengths, but increases with longer

network lengths. As a result, a larger volume flow is transported over longer distances.

This increases the average pipe diameter and resulting in higher pipe investment costs as

well ass increased heat losses. On the contrary, in the scenario where the nominal consumer

power decreases, higher volume flow rates are extracted at shorter network lengths, leading

to lower average pipe diameters. This has a positive impact on network distribution costs.

In contrast to that, the variations of the specific costs obtained from the scenarios with

3. Detailed Simulation Model

0 2 4 6 8 10

Branch length Lbra [km]

100

150

200

Inner pipe diameter din [mm]

Rising 1

Rising 2

Falling 2

Falling 1

Default

Figure 3.11: Inner pipe diameter as a function of the network length related to the largest

possible network expansion of 𝐿bra =10km for several variations of the linear distributions

of the nominal consumer power. Scenario: linear variation of the nominal consumer power.

randomly distributed nominal consumer power and randomly distributed branch lengths,

shown in Figures 3.10b and 3.10c, are relatively small. In both scenarios, the absolute

value of the relative difference in total distribution costs increases with an increased DOV

value. The maximum relative difference in total distribution costs is small compared to

the default model, being below (𝑐tot −𝑐tot,default)/𝑐tot,default =1.2%for the given example.

To summarize the investigation presented in this section, it has been shown that a

linear distribution of the LHD cannot be neglected when estimating the distribution costs

of a DCC. This is mainly affected by variability in the mean pipe diameter. However,

randomly distributed consumer power and branch length do not have a significant effect

on the total distribution costs of a DCC.

3.5 OFAT Sensitivity Study

The aim of the OFAT sensitivity study is to explore the impact of a single input parameter

and its sensitivity on the characteristic cost function. This helps to distinguish between

input parameters that have a significant effect on the distribution of the characteristic

function and those that have a weak impact. To achieve this objective, an OFAT sensitivity

analysis was conducted, as described in Section 3.2. The outcomes of this study are

presented in Section 3.5.1. Finally, a summary and a conclusion of the OFAT study are

given in Section 3.5.2.

3.5.1 Results of the OFAT Sensitivity Study

To evaluate the sensitivity study, two types of diagrams were selected. The first type of

diagram shows the characteristic function for each parameter when its default, minimum,

and maximum values are employed given in Table 3.2. This resulted in 26 individual

diagrams, which are illustrated in Figures 3.13a – 3.15e. Furthermore, the sensitivity

analysis was conducted at a branch length of 𝐿bra =10km using a heat map, as presented

in Figure 3.12. In this figure, each row displays the sensitivity of a particular input

parameter compared to the default model for the DCC. It should be noted that this

section only presents and discusses the results of the DCC, as the results obtained using

the SCC are qualitatively similar. The heat map for the SCC is provided in the appendix

and can be seen in Figure A.8.

To demonstrate how to interpret the heat map diagram, an example is presented below.

In Figure 3.12, the first two columns display the minimum and maximum percentage

changes of the respective input parameters with respect to their default values, which are

provided in Table 3.2. The remaining columns show the sensitivity of each component

3.5. OFAT Sensitivity Study

of the specific distribution costs (capital, pump, heat loss, and total) to the minimum or

maximum input parameter values. The results are presented in a visually clear manner by

categorizing the input parameters into those with low (±0...10%), medium (±10...40%),

high (±40...100%), or extreme (>±100%) sensitivity.

In the first row of Figure 3.12, the nominal power of each consumer 

𝑄con,n,𝑖,𝑗 was

reduced by 50% and increased by 100% relative to the default value. The reduction

resulted in an increase of capital LCOH of 38.2%, an increase of heat loss LCOH of

76.3%, and an increase of total LCOH of 33.9%. Doubling the nominal power of each

Input

value

min

Input

value

max

Capital

LCOH

min

Capital

LCOH

max

Pump

LCOH

min

Pump

LCOH

max

Q loss

LCOH

min

Q loss

LCOH

max

Total

LCOH

min

Total

LCOH

max

Qcon,n,i,j

τfull,con,i,j

ϑs,n

ϑamb

∆pn

pmax

∆pmin,sst

ξn

ηpump

τop

λins

λsoil

Ldep

∆Lcon,i,j

τinv

c∗

cth

ˆcinv,pump

ˆcinv,sst

-50 % +100 % +38.2 % -22.2 % +0 % -0 % +76.3 % -44.3 % +33.9 % -19.7 %

-50 % +100 % +100 % -50 % +0 % +0 % +100 % -50 % +73.4 % -36.7 %

-50 % +50 % +24.5 % -9.7 % +223.5 % -50.8 % -46.3 % +27.1 % +61.3 % -13.8 %

-50 % +50 % +0.1 % -0.1 % +0 % +0 % +8.9 % -8.9 % +1.4 % -1.4 %

-80 % +80 % -12 % +16.7 % -79 % +79 % +13.4 % -6.4 % -24 % +27.8 %

-90 % +900 % +0.1 % +2.6 % +0 % -0 % +0.1 % +2.7 % +0.1 % +1.9 %

-50 % +100 % -7.2 % +0 % +0 % +0 % +0 % +0 % -4.2 % +0 %

-62.5 % +56.2 % +43.2 % -14.4 % +0 % +0 % +0 % +0 % +25.1 % -8.4 %

-50 % +100 % -7.2 % +0 % -0.7 % +1.3 % +0 % +0 % -4.3 % +0.3 %

-100 % +100 % -7.2 % +0 % -16.4 % +16.4 % +0 % +0 % -8.2 % +4 %

-28.6 % +28.6 % +0 % +0 % +40 % -22.2 % +0 % +0 % +9.7 % -5.4 %

-50 % +0 % +0 % +0 % +0 % +0 % -50 % +0 % -7.6 % +0 %

-66.7 % +100 % -0.6 % +0.9 % +0 % +0 % -65.4 % +90 % -10.4 % +14.3 %

-83.3 % +100 % -0.2 % +0 % +0 % +0 % -19.9 % +2.6 % -3.1 % +0.4 %

-50 % +50 % +0.2 % -0.1 % +0 % +0 % +21.2 % -14 % +3.4 % -2.2 %

-50 % +100 % +0 % -0 % +0 % +0 % +0.6 % -0.8 % +0.1 % -0.1 %

-66.7 % +200 % -13.3 % +18.4 % -0.1 % +0.3 % -13.6 % +40.7 % -9.8 % +17 %

-50 % +50 % -21.1 % +21.1 % +0 % +0 % +0 % +0 % -12.3 % +12.3 %

-50 % +100 % -22.8 % +53.6 % +0 % +0 % +0 % +0 % -13.3 % +31.2 %

-50 % +33.3 % +55.5 % -12.6 % +0 % +0 % +0 % +0 % +32.3 % -7.3 %

-50 % +100 % +0 % +0 % -50 % +100 % +0 % +0 % -12.1 % +24.2 %

-50 % +100 % +0 % +0 % +0 % +0 % -50 % +100 % -7.5 % +15 %

-50 % +100 % -18 % +36 % +0 % +0 % +0 % +0 % -10.5 % +20.9 %

-50 % +100 % -3.9 % +7.8 % +0 % +0 % +0 % +0 % -2.3 % +4.5 %

<−100%

Extreme

−100% − −40%

High

−40% − −10%

Medium

−2% − −10%

Low

−2% −2%

None

2% −10%

Low

10% −40%

Medium

40% −100%

High

>100%

Extreme

Figure 3.12: Results of the sensitivity study using the data obtained from the OFAT

parameter study. The data was derived from evaluation the distribution costs at a network

expansion of 𝐿bra =10km assuming a DCC.

3. Detailed Simulation Model

consumer resulted in a 22.2%reduction of capital LCOH, a reduction of heat loss LCOH

of 44.3%, and a reduction of total LCOH by 19.7%. However, the pumping LCOH was

not affected by either an increase or a decrease in the nominal power of each consumer.

Nominal Consumer Power 

𝑄con,n,𝑖,𝑗

The nominal consumer power has a significant impact on the heat sold to the customer,

and consequently, on the LHD. However, increasing the nominal consumer power leads

to higher investment costs and heat losses due to the increased nominal pipe diameters.

This effect is more than compensated for by the higher share of heat sold. A reduction in

the nominal consumer power by 50%leads to an increase in the total LCOH by 33.9%.

On the other hand, doubling the nominal consumer power results in a reduction of the

total LCOH by 19.7%. Both capital and heat loss LCOHs are affected, while the pumping

LCOH remains unchanged. This is because an increase in the nominal consumer power

leads to higher volume flow rates and, thus, higher pump energy demand, which balances

out the reduction in LCOH resulting from increased heat sold to the customer.

The characteristic function distribution, shown in Figure 3.13a, demonstrates that the

nominal consumer power affects the total LCOH throughout the entire distribution path.

The three cases are shifted approximately in parallel, indicating that linear DHNs with

low values of LHD tend to be more expensive. However, even at low LHDs, linear DHNs

can be economically feasible if the network expansion can be reduced.

Thus, the nominal consumer power or, respectively, the LHD, exhibits a strong sensi-

tivity to the total LCOH. This sensitivity decreases significantly for higher values of the

nominal consumer power.

Consumer Full Load Hours 𝜏full,con,𝑖,𝑗

If the full load hours are reduced by half, the total LCOH is increased by 73.4%. Con-

versely, doubling the full load hours of the consumers results in a decrease in the total

LCOH by 36.7%. Therefore, the consumer full load hours demonstrate a stronger sensi-

tivity to the total LCOH than the nominal consumer power, although both parameters

affect the LHD to the same extent. However, altering the full load hours alone has no

impact on piping diameter, pipe investment costs, or heat losses, provided that the con-

sumer power remains constant. Thus, the sensitivity observed is mainly affected by the

increased proportion of heat that can be sold and not a combined effect of changing pipe

diameters and additional heat that can be sold.

Observing the distribution of the characteristic functions in Figure 3.13b, an approx-

imately parallel shift of the characteristic function can be seen. This shift is comparable

to the shift observed for the nominal consumer power of the consumers (see Figure 3.13a),

but more pronounced. It should be noted that the LHD is similar for the nominal con-

sumer power shown in Figure 3.13a and for the full load hours presented in Figure 3.13b.

This finding indicates that considering the LHD alone may lead to different outcomes if

the economic feasibility of a district heating network is to be estimated. In this context,

the linear load density (kW/m) may serve as an additional evaluation parameter if the

overall distribution costs are to be estimated, as will be examined in subsequent chapters.

Nominal Supply Temperature 𝜗s,n

The nominal supply temperature has a significant impact on the heat losses and the heat

loss LCOH. It also affects the nominal temperature difference, which in turn influences

capital and pumping costs, as described in Equation (2.32). The nominal supply tem-

perature exhibits medium to high positive sensitivity regarding the heat loss LCOH, as

3.5. OFAT Sensitivity Study

shown in Figure 3.12. Increasing the nominal supply temperature results in higher heat

loss costs, and vice versa, due to the increased temperature difference between the network

and the environment. On the other hand, the nominal supply temperature has a very high

negative sensitivity regarding the pumping LCOH. This is because increasing the nominal

supply temperature leads to an increased temperature spread, which, in turn, decreases

the volume flow rate and the electrical demand for pumping. The capital LCOH exhibits

medium negative sensitivity and is affected by an increase in the nominal temperature

spread. This increase leads to smaller pipe diameters and hence, to lower capital costs.

The total LCOH exhibits medium to high negative sensitivity, where the sensitivities due

to capital and pumping costs overcome the sensitivity of the heat losses.

The characteristic function shown in Figure 3.13c exhibits a strong slope increase if

low values of the nominal supply temperature are used (as shown by the dark gray line).

This is mainly due to the increased energy consumption of the pump, resulting from higher

volume flow rates. Therefore, shorter network expansions are more feasible than longer

network expansions at low values of the nominal supply temperature. When higher values

of the nominal supply temperature are used, only a weak reduction in distribution costs

is observed.

Nominal Pressure Loss Δ𝑝n

The nominal pressure loss has a medium positive sensitivity with respect to the total

LCOH, as shown in Figure 3.12. Regarding the capital LCOH, a decrease of 12.0%is

observed if the nominal pressure loss is reduced by 80%compared to the default parameter,

while an increase of 16.7%is observed if the nominal pressure loss is increased by 80%.

This might seem surprising, as a higher nominal pressure loss usually leads to smaller pipe

diameters and hence lower pipe investment cost. While this is true for pipe investment

costs, another effect is observed when the nominal pressure loss is increased. As a higher

nominal pressure loss leads to increased pressure losses, the required differential pressure of

the pumping stations needs to be split over several pumping stations, resulting in higher

investment costs of the pumping stations, which overcompensates the effect of smaller

pipe diameters. Moreover, the pumping LCOH is significantly affected by this value, as

an increase in nominal pressure loss directly increases the energy consumption for pumping.

Additionally, the heat loss LCOH is affected by the nominal pressure loss, as a reduction

of this value leads to larger pipe diameters, which increases the area available for heat

losses and vice versa.

In Figure 3.13e, the characteristic functions illustrate the impact of increased numbers

of pumping stations. For smaller network expansions, higher nominal pressure losses

correspond to lower total costs, while this positive effect is reversed for larger network

expansions due to additional investment costs for additional pumping stations. This effect

is demonstrated in Figure 3.13e by the number of discontinuities, which represents the

number of additional pumping stations. At the maximum nominal pressure loss value

(light gray line), there are five discontinuities, while at the minimum value, there are zero.

Maximum Operating Pressure 𝑝max

The maximum operating pressure (MOP) shows a negative sensitivity with respect to the

capital LCOH. This is because the MOP directly affects the pressure difference that can

be used for pressure losses. Increasing the MOP reduces the number of required pumping

stations, which in turn reduces the investment and therefore the capital costs. This effect

is illustrated in Figure 3.13h, where it can be seen that more pumping stations are required

if the minimum MOP value is selected (dark gray line).

3. Detailed Simulation Model

Nominal Pressure Loss Factor of Fixtures 𝜉n

The factor affecting the nominal pressure loss of fixtures shows a positive sensitivity with

respect to pumping LCOH. An increase in this factor leads to an increase in pressure drop,

which in turn may require additional pumping stations, and may negatively affect the

capital LCOH. However, the observed sensitivity of this factor with respect to the total

LCOH is very low. This is evident from the distribution of the characteristic function

shown in Figure 3.13j.

Pump Eﬀiciency 𝜂pump

The eﬀiciency of pumps leads to a medium negative sensitivity with respect to the pumping

LCOH, as illustrated in Figure 3.12. However, this parameter has no significant impact on

the capital and heat loss LCOH. Regarding the total LCOH, only a low negative sensitivity

is observed. Figure 3.14a shows that the characteristic function indicates a change in slope

without a parallel shift. Thus, the impact of different pump eﬀiciencies becomes stronger

with longer network expansions.

Operational Hours 𝜏op

Only one additional parameter was observed for the operational hours parameter, besides

the default value, which represents continuous annual operation. Reducing the operational

hours by fifty percent leads to a fifty percent reduction in heat losses, with no impact on

the remaining shares of the total LCOH. However, this reduction in operational hours may

also affect pumping costs, as a decrease in operational hours would lead to a higher power

being transported through the network. This effect is not considered in the model due

to assumptions made in Sections 2.4.5 and 2.6.2. The effect on the pressure loss LCOH

is expected to be minimal due to two opposing effects occurring at reduced operational

hours. On one hand, the lower operation hours of the pump would reduce annual power

consumption. On the other hand, the increased energy consumption of the pumps during

operational hours, due to the unchanged nominal consumer load and full load hours, would

lead to higher volume flow rates and increase energy consumption. In a real-life scenario,

it is expected that there would be a higher annual energy consumption of the pump due to

the quadratic relation between volume flow rate and pressure loss, if the operation hours

of the network were reduced.

Figure 3.14b displays the effect of the reduction of the operational hours, showing a

parallel shift in the total LCOH related to the reduced annual heat loss costs.

Heat Conductivity Coeﬀicient of the Pipe Insulation 𝜆ins

This parameter exhibits a high positive sensitivity with respect to the heat loss LCOH, as

shown in Figure 3.12. Increasing the value of 𝜆ins leads to higher heat losses and therefore

increases the heat loss LCOH. However, this parameter has no effect on the pumping and

capital LCOH. The shape of the characteristic function shown in Figure 3.14c indicates a

parallel shift when this parameter is varied.

Connection Length of the Consumers Δ𝐿con,𝑖,𝑗

This parameter exhibits a low positive sensitivity to capital costs and a low to medium

sensitivity to heat loss LCOH as shown in Figure 3.12. The increase in connection pipe

length leads to higher investment costs and relative heat losses, since the heat consumption

3.5. OFAT Sensitivity Study

remains unchanged. Therefore, the heat loss LCOH is affected. The characteristic function

in Figure 3.14g displays a slight parallel shift in response to changes in this parameter.

Linear Pipe Cost Coeﬀicient 𝐼1

The linear pipe investment coeﬀicient 𝐼1is responsible for determining the investment

costs associated with the inner pipe diameter 𝑑in, and therefore only exhibits a medium

positive sensitivity with respect to the capital LCOH. This parameter has a similar effect

on the total LCOH (as seen in Figure 3.12). Changing this parameter results in a variation

of the slope of the characteristic function, as demonstrated in Figure 3.14h. Thus, larger

variations in the total LCOH can be observed with longer network expansions when there

is a variation in the linear pipe cost coeﬀicient.

Interest Rate IR and Investment Horizon 𝜏inv

The two parameters have a direct impact on the annuity and consequently on the capital

costs. However, they do not affect the pumping and heat loss costs. The interest rate

exhibits a medium positive sensitivity, while the investment horizon shows a medium

negative sensitivity on the total LCOH, as shown in Figure 3.12. The characteristic

functions for both parameters shown in Figures 3.14i and 3.14j demonstrate a parallel

shift. This implies that any change in these input parameters will cause the total costs to

vary independently from the network expansion.

Electrical Power Costs 𝑐∗el

This parameter exhibits a high positive sensitivity in the pumping costs, resulting in a

medium positive sensitivity in the total costs, as shown in Figure 3.12. This parameter

does not have any effect on the heat loss or capital costs. In terms of the distribution

of the characteristic function shown in Figure 3.15a, variations in this parameter lead to

changes in the slope. Therefore, the electrical power costs have a more significant impact

on the total costs as the network expands.

Specific Heat Costs 𝑐th

In contrast to the electrical power costs, the specific heat costs parameter exhibits a

positive sensitivity with respect to the heat losses (Figure 3.12). However, it does not

affect the capital and pumping costs. Regarding the total LCOH, this parameter has

a low to medium sensitivity. Moreover, the trend of the characteristic function (Figure

3.15b) shifts in parallel when changing the specific heat costs. Therefore, the network

expansion does not strengthen or weaken the influence of this parameter.

Specific Investment Costs of Pumping Stations 𝑐inv,pump and Substations

𝑐inv,sst

Each of these parameters has a positive sensitivity to investment costs. Heat loss and

pumping costs are not affected by either value (see Figure 3.12). Specifically, the specific

investment costs of pumping stations show a stronger sensitivity to the capital LCOH

than the specific investment costs of substations. Although the magnitudes used for the

parameter study for 𝑐inv,pump and 𝑐inv,sst are similar, the need for multiple pumping stations

amplifies the influence of the pumping station investment costs.

This effect is also visible in the distribution of the characteristic function shown in Fig-

ures 3.15c and 3.15d. The characteristic function shows a similar distribution for both pa-

3. Detailed Simulation Model

rameters until the first discontinuity is reached. From this point forward, the 𝑐inv,pump pa-

rameter shows a stronger influence than the 𝑐inv,sst parameter. Furthermore, the 𝑐inv,pump

parameter leads to a change in the slope of the characteristic function. Therefore, the

effects of this parameter on the total costs increase with increasing network expansions.

Usage of the Simultaneity Factor SF

The effects of the simultaneity factor according to the network expansion can be seen in

Figure 3.15e. Here, the distribution is shown if the simultaneity factor is considered during

the network design or not. If this parameter is not considered, the overall cost will be

higher because the pipe diameters will be increased due to a higher nominal pipe volume

flow rates. Since the SF depends on the number of consumers, the SF decreases as the

network length increases (see Equation (2.7)). This leads to stronger influences of this

parameter with larger network expansions.

Remaining Parameters

The average ambient temperature 𝜗amb, the pipe roughness 𝑟, the resting pressure 𝑝R, the

minimum pressure difference of the substations Δ𝑝min,sst, the heat conductivity coeﬀicient

of the soil 𝜆soil, and the pipe installation depth 𝐿dep are all found to have subordinate

significance in the context of the total LCOH.

The influence of the average ambient temperature on the heat loss costs is observed

to be low. Figure 3.12 shows a negative sensitivity with respect to the heat loss LCOH.

However, the impact of this parameter on the characteristic function of the total LCOH

is found to be negligible, as shown in Figure 3.13d.

The investigations conducted here show that pipe roughness has a negligible impact,

as shown in Figure 3.12 and 3.13f.

The resting pressure of a system, 𝑝R, has a significant impact only on the capital LCOH

due to its effect on the number of pumping stations. As the resting pressure increases,

more pumping stations are required because a lower difference between the maximum

operating pressure and the resting pressure can be utilized for pressure losses. This trend

can be observed in Figure 3.13g.

The minimum pressure difference of the substations Δ𝑝min,sst can be considered negli-

gible since it has minimal impact, as shown in Figures 3.12 and 3.13i.

The heat conductivity coeﬀicient of the soil 𝜆soil the insulation class IC, and the instal-

lation depth of the pipes 𝐿dep, exhibit negligible effects on the heat losses, capital cost, and

pumping cost of the system. Furthermore, the distribution of the characteristic function

remains insignificantly affected by these parameters, as shown in Figures 3.14d, 3.14e, and

3.14f.

3.5. OFAT Sensitivity Study

3.5.2 Summary and Conclusion of the OFAT Sensitivity Study

The investigations have shown that a single input parameter can effect the specific heat

distribution differently. The analysis has identified parameters with strong and negligible

effects on the total LCOH. The nominal supply temperature, nominal consumer power,

consumer full load hours, and linear heat density are parameters that show a strong influ-

ence. Parameters with medium sensitivity according to the total LCOH are the nominal

design pressure loss, the maximum operational pressure, the heat conductivity coeﬀicient

of the insulation, the pipeline length of the consumers, the linear pipe investment coef-

ficient, the interest rate, the investment horizon, the electrical power price, the specific

heat costs, and the specific investment costs of the pumping stations. On the other hand,

resting pressure, pressure loss factor of fixtures, pump eﬀiciency, annual operation hours

of the DHS, heat conductivity coeﬀicient of the soil, pipe insulation class, and specific

investment costs of the substations exhibit a weak sensitivity. The average ambient tem-

perature, pipe roughness, minimum pressure difference of the substations, and installation

depth of the pipes have entirely negligible effects.

Based on the investigations, two different effects on the shape of the characteristic

function were observed. Some parameters demonstrated a parallel shift of the character-

istic function, while others showed a variation in its slope. A parallel shift indicates that

the parameter affects the total LCOH independently of the network expansion, whereas

a change in the slope indicates that the effect of the parameter becomes stronger as the

network expands. Parameters that affect the slope of the characteristic function are those

that have a strong sensitivity according to the pumping LCOH, as it increases with net-

work expansion. These parameters include the nominal temperature difference, which is

affected by the nominal supply temperature, the nominal pressure loss, the maximum op-

erational pressure, the nominal pressure loss of fixtures, the pump eﬀiciency, the electrical

power price, and the specific investment costs of the pumping stations. Additionally, the

linear pipe investment coeﬀicient and the simultaneity factor exhibit a significant change

in the slope of the characteristic function.

Finally it can be noted that the OFAT parameter study provided insights into the

main drivers of the system when only a single parameter is varied. However, it does not

investigate the combined effects of multiple input parameters on the distribution costs.

Therefore, a more comprehensive analysis is carried out in Section 3.6.

3. Detailed Simulation Model

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(a) 

𝑄con,n,𝑖,𝑗

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(b) 𝜏con,n,𝑖,𝑗

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(d) 𝜗amb

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(e) Δ𝑝n

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(f) 𝑟

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(g) 𝑝R

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(h) 𝑝max

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(i) Δ𝑝min,sst

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(j) 𝜉n

Figure 3.13: Distribution of the total LCOH related to the network expansion. Data was

derived from the OFAT parameter study assuming a DCC part I. Black: default, dark

gray: minimum, light gray: maximum.

3.5. OFAT Sensitivity Study

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(a) 𝜂pump

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(b) 𝜏op

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(d) 𝜆soil

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(e) IC

0246810

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(f) 𝐿dep

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(g) Δ𝐿con,𝑖,𝑗

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(h) 𝐼1

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(i) IR

0246810

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(j) 𝜏inv

Figure 3.14: Distribution of the total LCOH related to the network expansion. Data was

derived from the OFAT parameter study assuming a DCC part II. Black: default, dark

gray: minimum, light gray: maximum.

3. Detailed Simulation Model

0 2 4 6 8 10

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(a) 𝑐∗el

0 2 4 6 8 10

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

(b) 𝑐th

0 2 4 6 8 10

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

0 2 4 6 8 10

Branch length Lbra [km]

0.0

2.5

5.0

Total LCOH

cdst [ct/kWh]

(d) 𝑐inv,sst

0 2 4 6 8 10

Branch length Lbra [km]

Total LCOH

cdst [ct/kWh]

True

False

(e) Use SF

Figure 3.15: Distribution of the total LCOH related to the network expansion. Data was

derived from the OFAT parameter study assuming a DCC part III. Black: default, dark

gray: minimum, light gray: maximum.

3.6. Monte Carlo Study

3.6 Monte Carlo Study

The Monte Carlo study aimed to investigate the influence of input parameters on dis-

tribution costs across the full spectrum of possible values. This approach allows for the

determination of which parameters affect distribution costs and in what way. This is in

contrast to the OFAT study, which considers only the variation of one parameter relative

to a reference case. As described in Reference [104], a typical Monte Carlo simulation

consists of three steps. First, a predictive model is established, and dependent and in-

dependent variables are identified. Second, a probability distribution of the independent

variables is defined. Third, simulations are conducted using a random set of indepen-

dent variables until enough results are obtained to generate a representative sample. The

obtained data is then analyzed, and the results are evaluated.

The first step of the study involved setting up a detailed model, as explained in Sec-

tion 3.1, and defining the input parameters, which are listed in Table 3.2. The dependent

variables in this study corresponded to four types of specific distribution costs, namely

capital, heat loss, pressure loss, and total costs. In the second step, the input parame-

ters were randomly varied using a uniform probability distribution, which assumes that

each input parameter has an equal chance of occurring within the defined minimum and

maximum range specified in Table 3.2. A uniform probability distribution was selected

because it was assumed that all input parameters are equally likely to be selected when

designing a distribution network. The number of simulations required to generate enough

data depends on the specific case, which will be discussed in Section 3.6.1.

The results of the Monte Carlo simulation were evaluated using three steps. First,

the distribution of the dependent variables, including the total, capital, heat loss, and

pumping LCOH, were analyzed according to their statistical distribution, as discussed in

Section 3.6.2. Second, the offset between the distribution costs using a SCC and a DCC

was analyzed and discussed in Section 3.6.3. Third, the correlation between the input

parameters and the dependent variables was analyzed in Section 3.6.4. In addition, two

types of regression models were developed and are being discussed in Sections 3.6.5 and

3.6.6. The first type of regression model investigated the possible regression models and

their accuracy when using only a single input parameter. The second type of regression

model investigated the possible regression models and their accuracy when using multiple

input parameters.

3.6.1 Suﬀicient Number of Cases

To determine a suﬀicient number of cases for Monte Carlo simulations, an evaluation mea-

sure needs to be defined first. The number of cases is considered suﬀicient if this evaluation

measure is reached or falls below a certain threshold. Monte Carlo simulations generate a

randomly distributed set of data that can be statistically described by its mean value 𝜇∗

and standard deviation 𝜎∗. The mean value represents the average of all cases, whereas the

standard deviation represents the variation of the data. Both values approach their ideal

value for an infinite number of cases. However, since infinite simulations are impossible, it

can be asked how many cases are needed to achieve a relative change in both values smaller

than a certain value. This relative change is referred to as the residuum RES, which can

be calculated using Equation (3.32). Here, 𝑋represents either the standard deviation or

the mean, RESXrepresents the residuum of the mean or the standard deviation, and the

index 𝑖represents the number of cases considered. A residuum value of RESX<10−4 is

considered suﬀicient in this study.

3. Detailed Simulation Model

RESX=|𝑋𝑖−1−𝑋𝑖|

𝑋𝑖(3.32)

0 2000 4000 6000 8000 10000

Number of cases n[-]

LCOH [ct/kWh]

µ∗σ∗

Figure 3.16: Evaluation of mean value 𝜇∗and standard deviation 𝜎∗with increasing cases

of the Monte Carlo simulation.

The evolution of the mean value (𝜇∗) and the standard deviation (𝜎∗) of the total

LCOH at a network expansion distance of 𝐿bra =10km is shown in Figure 3.16 across a

growing number of cases. Notably, both measures exhibit substantial fluctuations when

a limited number of cases is considered. However, it is observed that these fluctuations

remain constant as the number of cases increases.

0 2000 4000 6000 8000 10000

Number of cases n[-]

10−7

10−4

10−1

RESµ∗[-]

µ∗µ∗

100 RESµ∗= 10−4

(a) Mean value 𝜇∗

0 2000 4000 6000 8000 10000

Number of cases n[-]

10−6

10−3

100

RESσ∗[-]

σ∗σ∗100 RESσ∗= 10−4

(b) Standard deviation 𝜎∗

Figure 3.17: Development of the residual values for mean value and standard deviation

with an increasing number of cases, including a rolling average over one hundred cases.

The residuals corresponding to 𝜇∗and 𝜎∗are illustrated in Figures 3.17a and 3.17b.

These figures present both the actual residuals and the associated rolling mean values cal-

culated over one hundred cases (𝜇∗100,𝜎∗100). As the number of cases under consideration

increases, a continuous reduction in residual values is observed. The actual residual values

3.6. Monte Carlo Study

for 𝜇∗and 𝜎∗show pronounced variability, which is related to the 𝜇∗and 𝜎∗values asso-

ciated with each added case. When the introduction of an additional case results in only

slight modifications to the overall 𝜇∗and 𝜎∗values compared to the previous number of

cases, the residual at that specific step tends to be low and vice versa. In order to reduce

the inherent variability in the residuals, the utilization of rolling mean values has been em-

ployed. The defined threshold value of RES𝜇<10−4 is attained for the 100-rolling-average

when 𝑛>5409. Similarly, considering the 100-rolling-average of the standard deviation,

the threshold value is reached when 𝑛>9633. Therefore, 𝑛=10000can be considered a

suﬀicient number of cases.

3.6.2 Distribution of LCOH

In a first evaluation step of the Monte Carlo simulation, the LCOH (dependent variables)

are to be analyzed according their probability distribution. Therefore, the LCOH are

shown as several histogram plots in Figures 3.18a – 3.18d for two total network expansions

of 𝐿bra =0.5km and 𝐿bra =10.0km to represent a relatively short and a relatively long

expanded linear DHN of the DCC2. The mean value, the standard deviation, the minimum,

and the maximum values for each probability distribution are given in Table 3.8 for the

DCC and in Table A.11 for the SCC. Please note that in this section only the results

obtained from the DCC are being discussed, since the analysis of the results obtained

from the SCC leads to qualitatively similar results.

0 20 40 60 80 100

Total LCOH cdst [ct/kWh]

500

1000

1500

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(a) Total LCOH

0 10 20 30 40

Capital LCOH ccap,dst [ct/kWh]

500

1000

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(b) Capital LCOH

0 20 40 60

Heat Loss LCOH closs,dst [ct/kWh]

1000

2000

3000

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

0 5 10 15

Pumping LCOH cpump,dst [ct/kWh]

200

400

600

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(d) Pumping LCOH

Figure 3.18: Histograms showing LCOH components of the Monte Carlo simulation. The

plots show the distribution if the LCOH for a comparable short (𝐿bra =0.5km) and a

fully expanded distribution network (𝐿bra =10km) assuming a DCC.

Based on the histograms presented in Figure 3.18, it is evident that the distribution

of all LCOH values is non-normal. This non-symmetry of the probability distribution

with respect to the mean value characterizes the distribution as logarithmic normal3. This

2See Figures A.9 – A.9d for the histogram plots related to the results of the SCC.

3The logarithmic distribution becomes evident if the data is plotted using a logarithmic scale of th

abcissa axis as shown in Figures A.11 (DCC) and A.10 (SCC) given in the appendix.

3. Detailed Simulation Model

Table 3.8: Mean, standard deviation, minimum, and maximum value of of several costs

components for two different network expansion. The data was derived from Monte Carlo

parameter study with 𝑛=10000and assuming a DCC.

Entity 𝐿bra 𝜎∗𝜇∗Minimum Maximum

Unit [km] [ct/kWh] [ct/kWh] [ct/kWh] [ct/kWh]

Total LCOH 0.5 4.21 4.99 0.47 78.43

Total LCOH 10.0 6.66 10.56 1.37 100.96

Capital LCOH 0.5 2.58 3.17 0.36 31.15

Capital LCOH 10.0 4.37 6.06 0.69 45.58

Heat loss LCOH 0.5 2.02 1.48 0.04 49.08

Heat loss LCOH 10.0 2.53 1.88 0.05 60.38

Pumping LCOH 0.5 0.14 0.17 0.01 1.26

Pumping LCOH 10.0 2.22 2.44 0.07 19.45

asymmetric probability distribution implies that the specific distribution costs vary within

a similar range for most cases, but some cases result in significantly high distribution costs.

According to the results presented in Figure 3.18a, it can be observed that longer net-

work expansions have a significant influence on the total LCOH. As the network expansion

increases, the probability of higher total distribution costs also increases. The mean value

increases from 𝜇∗=5.0ct/kWh for the short to 𝜇∗=10.6ct/kWh for a large network ex-

pansion (+112%). Additionally, the standard deviation increases from 𝜇∗=4.2ct/kWh

for the short to 𝜇∗=6.7ct/kWh for a large network expansion (+59%). The increasing

mean value suggests that the distribution costs increase significantly as the network expan-

sion grows. This is also supported by the increasing minimum and maximum values of the

total LCOH shown in Table 3.8. Moreover, the increased standard deviation indicates that

the total cost becomes more uncertain with longer network expansions. This phenomenon

is influenced by model parameters that gain greater relevance as the network expands over

longer distances. In the case of shorter network expansions, fewer influential parameters

affecting heat distribution were observed, which is in line with the findings of the OFAT

parameter study discussed in Section 3.5. However, as the network expands further, more

influential parameters become relevant, leading to increased variability in the resulting

heat distribution costs. As a result, a wider range of specific total costs becomes possible

for the same set of input parameters.

Considering the capital LCOH shown in Figure 3.18b, a similar trend is observed as

with the total LCOH. As the network expands, higher capital LCOH values become more

likely. The increase in mean from the short to the long distribution network is 91%, which

is slightly smaller than that observed for the total LCOH. This implies that the capital

LCOH increases slightly less compared to the total LCOH for the same network expansion.

The standard deviation increases by about 69%, which is a bit stronger than the relative

increase observed for the total LCOH.

According to the histogram in Figure 3.18c, it appears that the impact of total net-

work expansion on the heat loss LCOH is relatively minor compared to the other LCOH

values. The mean value increases by approximately 25%and the standard deviation by

approximately 27%when comparing the smaller network expansion to the longer one.

This suggests that the heat loss LCOH is less responsive to changes in network expansion

compared to the total, capital, and pumping LCOH.

Based on the histogram presented in Figure 3.18d, a significant dependence on the

total network expansion can be observed for the pumping LCOH. The mean value for the

3.6. Monte Carlo Study

long network expansion is more than 14 times higher than the mean value for the short

network expansion. This is mainly due to the fact that the pressure loss scales linearly

with the network expansion. A similar trend can be seen for the standard deviation, which

is about 15 times larger for longer network expansions than for short ones.

3.6.3 Offset Between LCOH of SCC and DCC

In Section 3.4, the distribution of the characteristic function of the default model was

examined, where an offset was observed between the characteristic functions of the SCC

and DCC. The distribution costs of the SCC were found to have higher values than the

DCC. In this section, we aim to investigate this offset for the results of the Monte Carlo

simulation to gain a deeper insight into the magnitude of this offset when exploring the

full range of possible input parameters. Therefore, Figure 3.19a displays the probability

distribution of the relative offset between SCC and DCC for two branch lengths, which

correspond to a relatively short and a relatively long network expansion. Additionally,

Figure 3.19b presents the mean values 𝜇∗and standard deviations 𝜎∗for several network

expansions.

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175

X[-]

100

200

300

Number of cases [-]

Lbra = 10.0 km

Lbra = 0.5 km

(a) Histogram

0.1 0.5 1.0 2.5 5.0 10.0

Branch length Lbra [km]

0.000

0.025

0.050

0.075

0.100

0.125

µ∗(X)

µ∗σ∗

0.00

0.01

0.02

0.03

σ∗(X)

(b) 𝜇∗and 𝜎∗according network expansion.

Figure 3.19: Relative offset between the total LCOH assuming a SCC and a DCC for

several network expansions 𝐿bra according to 𝑋=(𝑐dst,scc −𝑐dst,dcc)/𝑐dst,scc.

Figure 3.19a illustrates that the relative offset between SCC and DCC follows a normal

distribution. Increasing the network expansion from 𝐿bra = 0.5km to 𝐿bra = 10.0km

results in a slight increase of the mean value, indicating a corresponding increase in the

relative offset for longer network expansions. This observation is consistent with the

results from the default model, which showed a wider spread between the characteristic

function of the SCC and DCC for longer network expansions (see Figure 3.7 in Section

3.4.1). Figure 3.19b indicates that the mean value increases by approximately 𝜇∗=8.5%

for a network expansion of 𝐿bra =2.5km. Subsequently, the mean value remains constant

with further network expansions.

In addition to the change in mean value, the standard deviation also displays variation

with increasing network expansion. This is evidenced by the wider probability distribution

in Figure 3.19a for the longer network expansion. This finding is supported by Figure 3.19b,

which demonstrates a continuously increasing standard deviation with network expansion.

This increase in standard deviation suggests greater variability between the characteristic

functions of SCC and DCC with increasing network expansion.

These investigations show a significant offset between the distribution costs for SCC

and DCC. This offset ranges from 0...20%for all cases investigated in the Monte Carlo

simulation. The relative offset between SCC and DCC has a mean value of 𝜇∗=8.5%

with a standard deviation of 𝜎∗=2.9%across all network expansions.

3. Detailed Simulation Model

3.6.4 Correlation of Input Parameters and LCOH

To get a better on how the input parameters of the Monte Carlo simulation are related

to the LCOH, the correlations between the input parameters (independent variables) and

the LCOH (dependent variables) were analyzed. The correlation of two values can be

described by the Pearson correlation coeﬀicient 𝜌, which is defined according Equation

(3.33) [105]. Here, cov(𝑋,𝑌)represents the covariance of two random variables 𝑋and

𝑌.𝜎∗Xand 𝜎∗Yis the standard deviation of the two observed random variables 𝑋and 𝑌.

The correlation coeﬀicient is in a range of 𝜌=−1...1, where 𝜌=−1means that 𝑋and

𝑌are perfectly negative correlated and 𝜌=1means that 𝑋and 𝑌are perfectly positive

correlated. A correlation factor of 𝜌=0means that 𝑋and 𝑌are not correlated.

𝜌=cov (𝑋,𝑌)

𝜎∗X𝜎∗Y[−1≤𝜌≤1] (3.33)

In Figure 3.20 the correlation between the four LCOH values and all input parameters

are given for the results of the Monte Carlo simulation of a DCC4. Here, the branch length

𝐿bra is considered as independent parameter too. This was obtained by reading the LCOH

values of each Monte Carlo run at a randomly selected network expansion. Additionally,

the linear heat density 𝑞lin, the linear load density 𝑞lin and the annuity 𝑎are shown, which

were derived from the input parameters.

Qcon,n,i,j

τfull,con,i,j

ϑs,n

∆Lcon,i,j

∆pn

τinv

c∗

cth

pmax

cinv,pump

λins

ϑamb

∆pmin,sst

ξn

ηpump

λsoil

Ldep

cinv,sst

Lbra

qlin

Total

Capital

Heat loss

Pumping

-0.5 -0.44 0.1 -0.11 0.11 0.11 -0.11 0.13 0.1 0.1 -0.07 0.08 0.13 -0.02 0.01 0.02 0 0.03 -0.05 0.02 0 0.04 0.3 -0.58 -0.5 0.17

-0.48 -0.5 0.15 -0.1 0.1 0.05 -0.18 0.21 0.02 0 -0.11 0.13 0 0 0.01 0.03 0 0.02 -0.01 0 0 0.04 0.24 -0.6 -0.48 0.28

-0.46 -0.29 0 0.16 0.11 -0.04 0 -0.01 0.02 0.25 0 0.01 0.3 -0.04 0 0 0 -0.01 0 0.04 0.02 0.01 0.06 -0.47 -0.46 0

-0.01 0 0.01 -0.44 0 0.37 0 0 0.29 0 0 0 0 0 0.01 0.02 0 0.09 -0.15 0.02 -0.02 0.03 0.47 -0.01 -0.01 0

−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

Figure 3.20: Heat map of the Pearson correlation coeﬀicient for several components of the

distribution costs (total, capital, heat loss, pumping) and input parameters. Data was

derived from the Monte Carlo parameter study assuming a DCC.

Considering the total LCOH in Figure 3.20, correlation coeﬀicients between |𝜌| =

0.0...0.58 were observed. Among all input parameters, the consumer nominal power

𝑄con,n,𝑖,𝑗 and the consumer full load hours 𝜏full,con,𝑖,𝑗 exhibited the strongest correlation,

with correlation coeﬀicients of 𝜌=−0.5...−0.44. The negative correlation coeﬀicient for

both parameters implies that reducing either the consumer full load hours or the consumer

nominal power would result in increased distribution costs. The LHD, which was derived

from the consumer full load hours and the nominal consumer power using Equation (1.2),

showed the strongest correlation with a correlation coeﬀicient of 𝜌=−0.58. The correla-

tion coeﬀicient of the linear load density 𝑞lin was similar to that of the consumer power,

as the length of each branch segment was held constant, allowing the linear load density

to be directly derived from the consumer power (see Equation (1.3)).

The network expansion 𝐿bra exhibited the next strongest correlation, with a correlation

coeﬀicient of 𝜌=0.3. This finding indicates that distribution costs increase as the network

length increases, consistent with previous analyses. When the consumer nominal power

and the consumer full load hours are treated as a single input parameter expressed by the

4See Figure A.12 for a correlation heat map related to the SCC.

3.6. Monte Carlo Study

LHD, the network length exhibits the second strongest correlation coeﬀicient. Although

approximately half as strong as the correlation with the LHD, the correlation of network

length with distribution costs is non-negligible in determining specific distribution costs.

The next strongest correlation coeﬀicient is observed in the annuity factor 𝑎, which is

derived from the interest rate IR and the investment horizon, 𝜏inv, according to Equation

(2.37). The annuity exhibits a correlation coeﬀicient of 𝜌 = 0.17, indicating a weaker

correlation with the total LCOH than the LHD and the network expansion. The heat

conductivity coeﬀicient of the insulation 𝜆ins shows a weak correlation coeﬀicient of 𝜌=

0.13. The length of the connection pipes Δ𝐿con,𝑖,𝑗 and the nominal pressure loss Δ𝑝n

exhibit a medium positive correlation of 𝜌=0.11. The nominal supply temperature 𝜗s,n

shows a medium negative correlation coeﬀicient with the total LCOH of 𝜌=−0.11, which

is mainly affected by the relation between the nominal supply temperature and the nominal

temperature difference introduced in Section 2.4.7. The heat costs 𝑐th, the electricity costs

𝑐el, and the linear pipe investment coeﬀicient 𝐼1are characterized by a weak correlation

of 𝜌=0.1with the total LCOH. Finally, all remaining input parameters exhibit very low

correlation coeﬀicients with the total LCOH and are not further discussed in this section.

Considering the correlation coeﬀicient of the capital LCOH shown in Figure 3.20, the

result is very similar to the correlation coeﬀicients observed at the total LCOH. Parameters

that mainly or exclusively influence the capital costs show stronger correlation coeﬀicients

compared to the total LCOH. These parameters are 𝐼1,𝜏inv, IR, and 𝑎.

Based on the heat loss LCOH shown in Figure 3.20, it is observed that the input

parameters 𝜗s,n,𝑐th, and 𝜆ins exhibit a much stronger correlation compared to their corre-

lation with the total LCOH. This is mainly due to the fact that these parameters directly

influence the heat loss LCOH. The heat loss LCOH shows almost no correlation with the

network length 𝐿bra, which indicates that increasing network length does not necessarily

translate to higher heat loss. Additionally, the correlation of the LHD with heat loss

LCOH is significantly weaker compared to its correlation with the total LCOH.

The analysis of the pumping LCOH showed that only a few input parameters are

significantly correlated with it. These parameters include 𝜗s,n,𝑐∗el,𝜂pump, and 𝐿bra. While

a weak correlation exists for 𝜉n, it is not statistically significant.

In summary, the main drivers influencing the distribution costs along the full range of

observed input parameters are the LHD 𝑞lin and the liner load density 𝑞lin, which are both

strongly correlated to each other, the network expansion 𝐿bra, and the annuity factor 𝑎.

Additionally, the linear pipe cost coeﬀicient 𝐼1, the nominal supply temperature 𝜗s,n, the

heat costs 𝑐th, the electricity price 𝑐el, the length of the connection pipes Δ𝐿con,𝑖,𝑗, the

heat conductivity coeﬀicient of the pipe insulation 𝜆ins, and the nominal pressure loss Δ𝑝n

show a weak but statistically significant correlation on the distribution costs of a linear

DHN.

3.6.5 Estimation of the Distribution Costs Using Single Input

Regression Models

In this section, the applicability of regression models to predict distribution costs based

on a single independent input parameter using the data obtained from the Monte Carlo

study is analyzed. This approach eliminates the need for a detailed simulation. Two

regression models were considered: a linear model (Equation (3.34)) and a non-linear

model (Equation (3.35)). The independent variable, denoted as 𝑋, represents the input

parameters used for the Monte Carlo simulation. The coeﬀicients 𝐾1and 𝐾2represent

the coeﬀicients obtained from the models that fit the data.

3. Detailed Simulation Model

𝑐dst =𝐾1𝑋

[𝑋]+𝐾2(3.34)

𝑐dst =𝐾2

(𝑋

[𝑋])𝐾1(3.35)

In the first step, the eight main drivers identified in Section 3.6.4 that contribute to the

total LCOH are plotted as scatter plots in a 4x3 matrix presented in Figure 3.21 for a DCC5.

The distribution costs (total LCOH) are plotted against the respective input parameter.

It is worth noting that the ordinate (typical y-axis) has been limited to 𝑐dst <30ct/kWh,

which covers 99.0%of all cases. However, the complete dataset includes outlying values

up to 𝑐dst =92.6ct/kWh. The correlation between the LHD and the linear load density

is presented in the bottom right plot of Figure 3.21. The scatter plots for the remaining

input parameters are shown in Figures A.14 (DCC) and A.15 (SCC), but they are less

relevant in this section and therefore not further discussed.

The scatter plots of the relevant input parameters, shown in Figure 3.21, help visu-

alizing and understanding the data leading to the observed correlation coeﬀicients of the

total LCOH, as discussed in Section 3.6.4. The scatter plot in the upper left corner, show

the total LCOH as a function of the LHD, demonstrates a non-linear trend. Distribution

costs are higher for lower LHD, but significant variability is observed, particularly for low

values of the LHD (𝑞lin < 2.0MWh/m/a). Additionally, the results of the linear and

non-linear regression models, as well as their corresponding coeﬀicients of determination

𝑅2, are shown. The coeﬀicient of determination of the non-linear regression (𝑅2=0.64)

is the highest value among all considered input parameters, indicating that a non-linear

regression model utilizing the LHD as the independent variable is the most suitable pa-

rameter for estimating distribution costs. The corresponding model coeﬀicients, the linear

and non-linear models, as well as the RMSE and NRMSE, are provided in Table 3.9 for

aDCC6. The evaluation values of NRMSE =43.9%and 𝑅2=0.64suggest that using

the LHD with a non-linear model can replicate the distribution cost trend, but the results

are associated with a high degree of uncertainty. Nevertheless, the non-linear regression

model utilizing the LHD for estimating distribution costs aligns with estimations from

previous research (see Reference [12] p. 39). For this reason, this model is regarded as the

corresponding “reference model” for further analyses.

In the upper center plot of Figure 3.21, the non-linear relationship between the total

LCOH and the linear load density 𝑞lin is illustrated. However, this relationship exhibits

higher variability compared to that of the LHD, as indicated by a lower coeﬀicient of

determination value when compared to the non-linear regression of the LHD. By using

a non-linear model expressed by Equation (3.35), the relative error NRMSE increases

to 57.32%(refer to Table 3.9). Therefore, a greater level of uncertainty is observed in

comparison to the non-linear model employing the LHD.

Additionally, a strong correlation between 𝑞lin and 𝑞lin can be seen in the scatter plot

in the right lower corner of Figure 3.21. This is due to the fact that both values were

derived from the consumer power, but the LHD additionally includes variations of the full

load hours. Furthermore, it can be noted that the correlation of both values differs from

that of the LHD, which was obtained from the consumer full load hours.

The scatter plot shown in the upper right corner of Figure 3.21 represents the relation-

ship between network expansion 𝑐dst and distribution costs. It suggests a linear correlation

5See Figure 3.21 for the scatter plots related to a DCC.

6See Table A.12 for the correlation coeﬀicients and the reached error for a SCC.

3.6. Monte Carlo Study

Figure 3.21: Scatter plots including linear and non-linear regression of the relevant input

parameters for correlations coeﬀicients |𝜌|≥0.1. Data was derived from the Monte Carlo

parameter study assuming a DCC.

3. Detailed Simulation Model

between the two parameters indicating that the distribution costs are lower for shorter net-

work expansions. This observation is consistent with the characteristic function, which

shows a continuously growing trend for constant values of the LHD. However, the coeﬀi-

cient of determination for a linear regression model is very low (𝑅2=0.07). Moreover, the

relative error for a linear regression model using Equation (3.34) leads to a large relative

error of NRMSE =70.34%. Therefore, the network expansion 𝐿bra alone is not suitable

to estimate the distribution costs.

The remaining parameters, including the annuity 𝑎, the heat conductivity coeﬀicient

of the pipe insulation 𝜆ins, the nominal pressure loss Δ𝑝n, the consumer connection length

𝐿con,𝑖,𝑗, the nominal supply temperature 𝜗s,n, the heat costs 𝑐th, the linear pipe cost

coeﬀicient 𝐼1, and the price for electricity 𝑐el, also exhibit a linear relationship with the

distribution costs. However, their low coeﬀicients of determination suggest that these

parameters are not suitable for estimating the distribution costs on their own.

Table 3.9: Input parameters of the single input regression model according to Equations

(3.34) and (3.35) including reached accuracies and assuming a DCC.

Model

input 𝑋Type Unit of 𝑋 𝐾1𝐾2[ct/kWh] RMSE [ct/kWh] NRMSE [%]

𝑞lin Lin. [MWh/m/a] −3.0678ct/kWh 12.9997 4.6794 59.95

𝑞lin Non-lin. [MWh/m/a] 0.6456 8.0327 3.4260 43.89

𝑞lin Lin. [kW/m/a] −8.3245ct/kWh 13.4189 4.9715 63.69

𝑞lin Non-lin. [kW/m/a] 0.5847 5.0773 4.4736 57.32

𝐿bra Lin. [km] 0.5889ct/kWh 4.8641 5.4898 70.34

3.6.6 Estimation of the Distribution Costs Using Multiple Input

Regression Models

The results from the previous Section 3.6.5 have demonstrated that using only a single

input parameter to estimate distribution costs is insuﬀicient leads to poor accuracies. A

non-linear approach using the LHD with a non-linear model was used to achieve the best

accuracy of NRMSE =43.9%, which is still considered poor. To improve the prediction

accuracy and determine the extent to which it can be improved, a multi-dimensional linear

regression model was derived in the form of Equation (3.36). The non-linear approach was

necessary due to the non-linear relationship between the LHD and linear load density in

relation to distribution costs. The sum term in Equation (3.36) represents all remaining

input parameters, which were considered using a linear description. Depending on the

number of input parameters used, the model accuracy varies.

𝑐dst =𝐾0+𝐾1

(𝑞lin

MWh/m/a)𝐾2+𝐾3

(𝑞lin

kW/m)𝐾4+25

∑

𝑖=5𝐾𝑖𝑋𝑖

[𝑋𝑖](3.36)

The corresponding coeﬀicients for models using several input parameters are given in

Tables A.13 (DCC) and A.14 (SCC). The RMSE and NRMSE values obtained using several

input parameters using the DCC are shown in Figure 3.227. The order of parameters was

determined based on the correlation coeﬀicient obtained from the total LCOH, as shown

in Figure 3.20.

The nominal consumer power, the consumer full load hours, the investment horizon,

and the interest rate were considered based on the derived parameters of the annuity, the

7Please see Figure A.19 for the prediction accuracy for a SCC.

3.6. Monte Carlo Study

1234567822

(all)

Number of parameters

RMSE [ct/kWh]

NRMSE [%] and

number of coefficients [-]

Figure 3.22: RMSE and NRMSE of the multiple input regression model. The data was

obtained form the Monte Carlo simulation assuming a DCC. The vertical bars correspond

to the RMSE (left ordinate) and NRMSE (right ordinate). The line plot corresponds to

the number of coeﬀicients (right ordinate).

LHD, and the linear load density. The lowest relative error of NRMSE = 25.9%was

achieved when all input parameters were considered. This is the best possible accuracy

achievable using a multi-dimensional regression model based on the obtained data from

the Monte Carlo simulation and the selected model. Using different types of regression

functions is unlikely to increase accuracy significantly because the data shown in Figures

3.21 and A.14 indicate that, apart from the LHD and linear load density, just linear

relations between distribution costs and respective input parameters are to be expected.

The non-linear relations of the LHD and linear load density have already been taken into

account.

The number of model coeﬀicients varies between three if just the LHD is considered,

and 25 if all input parameters are considered. An increase in accuracy when using one

parameter (NRMSE = 43.9%) or two parameters (NRMSE = 42.1%) is minimal (see

Figure 3.22). This is because the first input parameter is the LHD, and the second

is the linear load density. As both parameters have a strong correlation (see plot at

bottom right corner in Figure 3.21), the amount of additional information gained by

adding the linear load density is relatively low. The next considered parameter is the

network expansion, which leads to a total number of three input parameters. Adding

this parameter provides additional information to the regression model and leads to a

decreased relative error of NRMSE =36.1%. The accuracy of the model continuously

increases as more parameters are taken into account, but this also leads to an increase

in model coeﬀicients that need to be considered, thereby increasing the complexity of the

regression model. The corresponding model coeﬀicients and the exact order of the model

input parameters are provided in Table A.13 in the appendix for various numbers of input

parameters.

3.6.7 Conclusion and Summary

The Monte Carlo study was employed to analyze the effects of a wide range of input pa-

rameters on distribution costs. The model investigations involved a total of 𝑛=10000

simulations, with residuals of the mean value and standard deviation below 3.0⋅10−5.

The results indicated that the distribution costs followed a logarithmic normal distribu-

tion, suggesting that the majority of the costs fell within a narrow range, but some outliers

occurred. This implies that certain combinations of input parameters can result in signif-

icantly higher distribution costs.

3. Detailed Simulation Model

Furthermore, an analysis of the correlation between the observed input parameters

and the distribution costs showed that four main drivers could be identified: the LHD,

the linear load density, the network expansion, and the annuity factor. Additionally, other

input parameters were found to have a low but existing correlation value with the specific

distribution costs. The input parameters LHD and the linear load density exhibited a non-

linear relation to the distribution costs, while all other input parameters showed a linear

relation to the distribution costs. It was also estimated that a SCC results in greater

distribution costs, with an average offset between the distribution costs using a SCC or a

DCC of approximately 8.5%.

If only a single input parameter is used to estimate the distribution costs using a

regression model, a non-linear model using the LHD as independent variable should be

used. The resulting model shows an average relative error of NRMSE =43.9%. However,

using another single input parameter leads to much worse results.

For this reason, multiple input regression models were investigated, resulting in a

decrease in relative estimation error to NRMSE = 25.9%. This value represents the

maximum possible accuracy of a regression model, as determined by the results of the

Monte Carlo simulations. It is less than one third of the relative error obtained when

using only the LHD as the input parameter for distribution cost estimation. However, the

average model error of NRMSE =25.9%corresponds to an absolute error of RMSE =

2.02ct/kWh, which is acceptable for rough estimations of distribution costs but insuﬀicient

for making robust and decisive decisions regarding the suitability of an area for district

heating supply.

4 Analytical Model

The investigations conducted in Sections 3.6.5 and 3.6.6 utilizing a regression model to

estimate heat distribution costs have shown that while this method exhibits reduced com-

plexity, its prediction accuracy is limited to approximately 25%. In light of the main

objective of this thesis1, this level of accuracy may remains unsatisfactory. The reason is

that the low accuracy can result in unreliable assessments, particularly during early plan-

ning stages, leading to cost risks and potential areas being excluded from district heating

supply. Consequently, if higher prediction accuracy is desired, regression models are not

suitable for estimating heat distribution costs.

In order to enhance prediction accuracy, it is necessary to conduct an investigation of

the individual DHN through detailed network analysis, typically accomplished by means

of a detailed network simulation. One approach to conducting such an investigation is by

employing a simulation tool similar to the detailed model presented in Chapter 3. However,

it should be noted that these detailed models have a significant drawback, particularly

during the early stages of planning, as the creation of a simulation is a time-consuming

process and necessitates a certain level of user expertise.

The detailed model introduced in Chapter 3 primarily served the purpose of solving

the techno-economic system equations outlined in Section 2.6.2. Attempting to solve

these equations manually would yield in a considerable number of computational steps

and is thus not advisable. This intensive computational effort arises from the need for

discretization of the network branches when considering a DCC. Moreover, each consumer

within the network possesses unique parameters (such as nominal connection power and full

load hours), which can significantly impact the network costs when expanding the network.

Additionally, the developed detailed model involved a high number of computational steps,

as an iterative approach was employed to determine the pipe friction factor, and linear a

interpolation was utilized to estimate the insulation thickness for each pipe segment.

However, in order to achieve a higher level of accuracy in estimating heat distribution

costs, exceeding that of regression models while requiring significantly less computational

effort than the detailed model, a novel method has been developed. This method aims to

minimize the number of computational steps to a bare minimum, enabling a wide range of

users to utilize it effectively. Consequently, a precise estimation of distribution costs can

even be accomplished through manual calculations, obviating the necessity for a computer

program and facilitating method utilization.

This chapter introduces the analytical model, which employs the mentioned method-

ology. The methodology utilized is described in detail, and the algorithm is presented in

Section 4.1. The prediction accuracy of the developed model is subsequently assessed in

comparison to the detailed simulation model in Sections 4.2 and 4.3. Section 4.2 focuses on

comparing the cost distribution along the network expansion of the default model, while

Section 4.3 compares the distribution costs for a range of possible input parameters using

the data retrieved of the Monte Carlo simulation presented in Section 3.6.

4.1 Description of the Methodology

The primary objective of the analytical model is to develop an algorithm that enables the

identification of a functional relationship between the distribution costs and the network

1To develop a method that enables the estimation of heat distribution costs with enhanced accuracy

and reduced effort.

4. Analytical Model

expansion of a linear DHN, according to Equation (4.1). The aim is to minimize the

number of required computation steps when using this algorithm.

𝑐dst =𝑓(𝐿bra,input parameters)(4.1)

The foundation for establishing the functional relationship represented by Equation

(4.1) is provided by the formulations of the distribution costs outlined in Section 2.6.2.

These system equations are applied to a simplified representation of a linear DHN, as

shown in Figure 4.1. This model description incorporates certain simplifications compared

to a real-world linear DHN application. The analytical model comprises a single heat

generator that supplies multiple consumers through a main branch (see Figure 4.1). Each

consumer is connected to the main branch through a connection pipe at a node, and

multiple consumers can branch off at each node, denoted by the parameter 𝑛rad. In

contrast to a real linear DHN application, the network structure of the analytical model

assumes an equidistant distribution network for both the branch and connection pipes.

Therefore, multiple consumers from the real DHN application are consolidated to conform

to this assumption. Consequently, the network structure of the analytical model can be

characterized by the average length parameters Δ𝐿bra and Δ𝐿con, along with the number

of consumers at each node 𝑛rad.

Main branch

Connection

Consumer parameters:

Node

Reality:

Analytical model:

Consumer

parameters:

Legend:

Consumer

Node

Pipe

Heat generator

4..

2 3 4

Figure 4.1: Model structure and graphical explanation of the corresponding linear DHN

of the analytical model. The analytical model simplifies the topological structure by using

equally sized network segments.

In a real linear DHN, each consumer can be described by unique characteristics, such as

a custom nominal connection power and full load hours. However, in the analytical model,

these descriptions have been simplified by employing an average value of the consumer full

load hours 𝜏full,con, which is assumed to be valid for all consumers. As discussed in Section

3.4.2, the distribution of the LHD along the network expansion can significantly impact

the distribution costs. Therefore, the consumer nominal power is not estimated based on

an averaged value but rather described by a linear function of the LHD, represented by

the coeﬀicients 𝑞∗

0and 𝑞∗

1. The relationship between the consumer nominal power and the

LHD is discussed in detail in Section 4.1.3. For now, it is important to note that the

heat demand of each consumer in the analytical model can be expressed based on the

branch length 𝐿bra, the average full load hours 𝜏full,con, and the coeﬀicients 𝑞∗

0and 𝑞∗

1. All

4.1. Description of the Methodology

remaining technical and economic input parameters utilized by the analytical model are

given in Table 4.1. Some of these input parameters may differ from those in the detailed

model, as explained further below.

Table 4.1: Overview of technical and economical model parameter of the analytical model.

Technical parameters Economical parameters

𝑛rad;𝜗s,n;Δ𝜗n;𝜗amb;𝜁pump;Δ𝑝n;𝑝R;𝜉n;

Δ𝑝min,sst;𝑝max;𝜏op;𝜆ins;𝜆soil;𝐿dep;𝐷0;

𝐷1;𝐷2;𝑓∗0;𝑓∗1;

𝐼0;𝐼1;𝐼𝑅;𝜏inv;𝑐OM,var,dst;𝑐OM,fix,dst;𝑐∗

el;

𝑐th;𝑐inv,pump;𝑐inv,sst

In order to obtain an analytical solution, it is necessary to establish a representation

of the distribution costs that aligns with the model description shown in Figure 4.1. As

discussed in Section 2.6.2, the distribution costs of a generic DHN consists of capital costs,

pumping costs, heat loss costs, and O&M costs (refer to Equation (2.40)). When applied

to the analytical model, the distribution cost structure illustrated in Figure 4.2 can be

derived. According to this figure, the capital costs in the analytical model can be further

classified into capital costs associated with the connection and branch pipes, as well as

capital costs related to the consumer substations and pumping stations. The heat loss

costs can be subdivided into heat loss costs of the branch and connection pipes. In total,

eight cost components are required to estimate the overall distribution costs, which are

denoted by a diamond symbol in Figure 4.2.

♦

Figure 4.2: Composition of the heat distribution costs of the analytical model. Cost

components marked by a diamond need to be provided by the analytical model.

In order to obtain an analytical solution, several adjustments and transformations of

the system equations compared to the detailed model were necessary. Initially, an explicit

formulation of the pipe friction factor and diameter was required to enable the calculation

of pipe diameters without the need for iterative methods. This aspect will be further

discussed in Section 4.1.1. Next, a continuous formulation of the pipe insulation thickness

was developed to address the limitations of the linear interpolation utilized in the detailed

model. This will be explained and discussed in Section 4.1.2. Subsequently, it was essential

to provide a comprehensive description of the LHD for the entire network. This description

will be expressed in relation to the input parameters 𝑞∗

0and 𝑞∗

1of the analytical model

which will be explained and discussed in Section 4.1.3. Furthermore, the formulation of

an average pipe diameter was required to estimate the distribution costs using a DCC.

A detailed explanation of this estimation will be presented in Section 4.1.4. Lastly, the

developed algorithm for calculating the distribution costs using the analytical model will

be introduced and discussed in Section 4.1.5.

4. Analytical Model

4.1.1 Formulation of the Pipe Friction Factor

In the detailed model, the determination of pipe diameter for each pipe segment involved

solving Equations (3.7) and (3.8) iteratively. This was necessary due to the dependence of

the pipe friction factor 𝑓∗on the inner pipe diameter 𝑑in (see (2.21)). However, employing

this approach in the analytical model is not ideal as it requires a significant number of

computational steps. Consequently, an explicit formulation of the pipe friction factor was

employed, as described by Equation (4.2). This formulation only requires the volume

flow rate and two regression parameters, 𝑓∗

0and 𝑓∗

1. The following sections provides the

derivation of this formulation and the determination of the regression parameters.

𝑓∗=( 

𝑉

m3/s)𝑓∗

1𝑒𝑓∗

0(4.2)

The parameters affecting the pipe friction factor can be mathematically represented by

Equation (4.3). Generally, the pipe friction factor depends on the Reynolds number Re,

the pipe roughness 𝑟, and the inner pipe diameter 𝑑in (see Equation (2.21)). The Reynolds

number itself depends upon the dynamic viscosity 𝜇, the average flow velocity 𝑣, and the

inner pipe diameter 𝑑in. The dynamic viscosity characterizes the material property of the

used heat transfer fluid and is influenced by the medium, a reference temperature 𝜗ref,

and a reference pressure 𝑝ref. The average flow velocity can be calculated from the inner

pipe diameter and the volume flow rate 

𝑉using the principle of continuity. According to

Equation (4.4), the inner pipe diameter depends on the volume flow rate 

𝑉, the density 𝜚,

the nominal pressure loss Δ𝑝n, and the pipe friction factor 𝑓∗. The density is dependent on

the medium, reference pressure, and reference temperature. Consequently, the inner pipe

diameter can be expressed as a function of the volume flow rate, reference temperature,

reference pressure, pipe friction factor, and nominal pressure loss (refer to Equation (4.4)).

Thus, the pipe friction factor 𝑓∗can be expressed as a functional relationship using the

medium, volume flow rate, inner pipe diameter, pipe roughness, reference temperature,

reference pressure, and nominal pressure loss.

𝑓∗=𝑓(Re,𝑑in,𝑟)=𝑓(𝜇,𝑣,𝑑in,𝑟)=𝑓(Medium,

𝑉,𝜗ref,𝑝ref,Δ𝑝n,𝑟) (4.3)

𝑑5

in =8

𝑉2𝜚(Medium,𝜗ref,𝑝ref)

𝜋2Δ𝑝n𝑓∗(4.4)

In the subsequent step, the iterative approach was employed to investigate the dis-

tribution of the friction factor and determine the sensitivity with respect to the input

parameters. To accomplish this, an OFAT parameter study was conducted, where the

friction factor corresponding to various parameter values was analyzed. In this context,

a default model was defined, and the input parameters were systematically varied within

their minimum and maximum ranges. For this particular case study, a volume flow rate

ranging from 

𝑉=2.43⋅10−3...0.243m3/s was considered. This range corresponds to a nom-

inal thermal load of 

𝑄=0.1...10MW at a nominal temperature difference of Δ𝜗n=10°C.

Thus, it covers the range applicable to most practical DH applications. The selected heat

transfer medium was water, utilizing the IAPWS-IF97 formulation [96]. The default, mini-

mum, and maximum values of the remaining parameters 𝑟,𝜗ref,𝑝ref, and Δ𝑝n, were varied

according to the specified values presented in Table 3.2. Here, 𝜗ref corresponds to the

values assigned to 𝜗s,n, and 𝑝ref corresponds to the values given for 𝑝R.

The results of the OFAT case study are shown in Figure 4.3, illustrating the relationship

between the pipe friction factor and the volume flow rate. Across all investigated scenarios,

a consistent pattern of monotonous decrease is observed as the volume flow rates increase.

4.1. Description of the Methodology

Moreover, most scenarios show trends closely aligned with the default case, indicating that

the influence of these parameters remains small within the designated parameter range.

The most significant deviation from the default case is observed in the 𝑟max scenario,

primarily related to the increased pipe roughness. In contrast, the 𝑟min scenario shows no

significant deviation from the default scenario. Furthermore, the parameters 𝜗ref and Δ𝑝n

show slight but observable differences compared to the default model. However, there is

no significant variation in the reference pressure compared to the default model, indicating

that its impact can be neglected.

0.00 0.05 0.10 0.15 0.20

Volume flow rate ˙

V[m³/s]

0.0100

0.0125

0.0150

0.0175

0.0200

Pipe friction factor f∗[−]

Default

rmin

rmax

∆pn,min

∆pn,max

ϑn,min

ϑn,max

pn,min

pn,max

Figure 4.3: Pipe friction factor related to the volume flow rate for several sets of input

parameters.

The functional relationship between the pipe friction factor 𝑓∗and the volume flow

rate 

𝑉can be represented by a logarithmic regression function as described in Equation

(4.2). To assess the accuracy of this regression function comprehensively, a Monte Carlo

simulation with 𝑛=1000cases was conducted, randomly varying the relevant input pa-

rameters 𝑟,𝜗ref, and Δ𝑝n. The results were evaluated by comparing the actual distribution

of the function 𝑓∗=𝑓( 

𝑉)to the distribution obtained from the regression function.

Across all cases, an average relative error of NRMSE =0.38%with a standard devi-

ation of 𝜎∗=0.27%was observed. The maximum error encountered was NRMSEmax =

1.01%. These results indicate that the utilization of Equation (4.2) yields highly accurate

results when approximating the relationship between 𝑓∗and 

𝑉. Furthermore, the input

parameters 𝑟,𝜗ref, and Δ𝑝ncan be considered constant for each DHN project. There-

fore, a specific set of correlation parameters 𝑓∗

0and 𝑓∗

1exists for each project, allowing a

continuous expression of the pipe friction factor in relation to the volume flow rate. The

corresponding values for 𝑓∗

0and 𝑓∗

1are provided in Table A.15 for a wide range of 𝑟,𝜗ref,

and Δ𝑝npermutations.

The investigations in this section have shown that the pipe roughness exhibits the high-

est sensitivity in relation to the pipe friction factor. Furthermore, the studies conducted in

Sections 3.5 and 3.6 have already demonstrated that the impact of pipe roughness on total

costs is relatively small. Therefore, variations in the pipe friction factor are also expected

to have a negligible effect on distribution costs. Consequently, the regression coeﬀicients

𝑓∗

0and 𝑓∗

1of the default model presented in Equation (4.5) can be used to describe the

pipe friction factor with suﬀicient accuracy.

𝑓∗

0=−4.7149and 𝑓∗

1=−8.6692⋅10−2 (4.5)

4.1.2 Formulation of the Pipes Insulation Thickness

The insulation thickness plays a crucial role in characterizing heat losses within a DHN. In

real DHN applications, the precise value of the insulation thickness is determined by the

4. Analytical Model

pipe manufacturer, who typically provides multiple insulation classes corresponding to dif-

ferent pipe sizes. In the detailed simulation model, the insulation thickness was obtained

directly from manufacturer data. When the pipe diameters were calculated continuously,

the insulation thickness was linearly interpolated using the manufacturer’s data. While

this approach is suitable for the detailed model, it introduces additional computational

steps that are not practical using the analytical model. Therefore, a continuous repre-

sentation of the insulation thickness was derived as a function of the inner pipe diameter

𝑑in.

Figure 4.4 illustrates the ratio between the outer insulation diameter 𝑑out,ins and the

inner insulation diameter 𝑑out,ins for a specific dataset obtained from a single KMR pipe

with three available insulation classes (IC). The data points were used to fit a regression

function represented by Equation (4.6).

𝑑out,ins

𝑑in,ins =𝜙= 𝐷0

(𝑑in

m)𝐷2+𝐷1(4.6)

0.05 0.10 0.15 0.20 0.25

Inner pipe diameter din [m]

dout,ins/din,ins [1]

IC 1

IC 1 fit

R²=0.985

IC 2

IC 2 fit

R²=0.992

IC 3

IC 3 fit

R²=0.994

Figure 4.4: Ratio of outer and inner insulation diameter related to the inner pipe diameter

valid for KMR-uno pipes. Data points were derived from Table A.8.

The results demonstrate a decreasing trend in the ratio of the outer and inner insulation

diameter as the inner pipe diameter increases. Additionally, an asymptotic behavior is

observed for larger values of 𝑑in. The coeﬀicient of determination 𝑅2for each insulation

class indicates an excellent fit to the data. Detailed information regarding the regression

parameters 𝐷0,𝐷1, and 𝐷2for individual pipes can be found in Table 4.2.

Table 4.2: Regression coeﬀicients to estimate the ratio of the inner and outer pipe insula-

tion diameter based on Equation (4.6) for single KMR, MMR and PMR pipes.

IC D0 D1 D2

KMR-uno

1 0.0464 1.2561 0.9873

2 0.02822 1.4814 1.1777

3 0.03050 1.6590 1.1931

MMR-uno 1 0.0214 1.3134 1.2132

PMR-uno 1 0.0615 1.1139 0.8644

2 0.0302 1.4590 1.0839

4.1. Description of the Methodology

4.1.3 Determination of the Network LHD

As discussed previously, the consumers in the analytical model were characterized by an

average value of their full load hours and a linear relationship between the LHD and the

network length. According to Equation (1.2), the LHD is defined as the ratio of the annual

heat demand 𝑄ato the corresponding network length 𝐿. The LHD can be formulated for

the entire distribution network, where the total annual heat demand of all consumers is

divided by the total network length. This value represents the average LHD of the entire

network and will be denoted as 𝑞lin,nw. Additionally, the LHD can be calculated for a local

segment of the network, where the cumulative annual heat demand of all consumers in that

segment is divided by the segment’s network length (including the length of connection

pipes). This approach allows a more specific consideration of varying heat densities along

the distribution network and will be referred to as the local LHD 𝑞lin,loc.

Reality:

Analytical

model:

Figure 4.5: Graphical explanation of the connection between the consumer heat demand,

the average LHD of the network, the local LHD, and its linear regression related to the

network expansion 𝐿bra.

In Figure 4.5, a simplified example is presented to illustrate the relationship between

consumer heat demands and the local LHD. At the top, a more realistic linear DHN is

presented, consisting of thirteen non-equally distributed consumers. The heat demand of

each consumer is represented by black dots in the diagram below. To estimate the local

LHD, the network is divided into several equally sized sections 𝑖, with five sections shown

in the example. For each section, the local LHD can be calculated using Equation (4.7). In

this equation, the numerator corresponds to the cumulative heat demand of all consumers

within the section, and it is divided by a reference length, which represents the total length

of the branch and connection pipes in that segment (as given by Equation (4.8)).

𝑞lin,loc,real,𝑖 =(∑𝑄con)𝑖

Δ𝐿ref

(4.7)

Δ𝐿ref =Δ𝐿bra +𝑛rad Δ𝐿con,a(4.8)

The resulting local LHD values 𝑞lin,𝑖 for each segment are plotted as circles in the diagram

of Figure 4.5. Since an equally sized network was used in this example, the derived local

LHD serves as a measure of heat density within each section. These data points can

be used to establish a linear regression function (represented by the black line in Figure

4.5) in the form of Equation (4.9), providing a continuous representation of the local

LHD. This regression function serves as the input function for the analytical model, with

4. Analytical Model

the coeﬀicients 𝑞∗

0and 𝑞∗

1specified as input parameters. Furthermore, each segment is

characterized by surrogate consumers positioned at the end of each section. Their annual

heat demand 𝑄con,a,𝑖,𝑗 and nominal connection power 

𝑄con,n,𝑖,𝑗 can be expressed using the

branch length 𝐿bra, the reference length Δ𝐿ref, and the coeﬀicients 𝑞∗

0and 𝑞∗

1as given by

Equations (4.10) and (4.11). This approach minimizes the consumer-related input data

while retaining a comprehensive consumer representation to avoid excessive simplification

of the model.

𝑞lin,loc,am,𝑖 =𝑞lin,loc,𝑖 =𝑞∗

1𝐿bra,𝑖+𝑞∗

0(4.9)

𝑄con,a,𝑖,𝑗 =𝑞lin,loc,𝑖Δ𝐿ref

𝑛rad =(𝑞∗

1𝐿bra,𝑖+𝑞∗

0)Δ𝐿ref

𝑛rad

(4.10)



𝑄con,n,𝑖,𝑗 =𝑄con,a,𝑖,𝑗

𝜏full,con =(𝑞∗

1𝐿bra,𝑖+𝑞∗

0)Δ𝐿ref

𝜏full,con 𝑛rad

(4.11)

Hence, the local LHD serves as a normalized measure for the consumer heat demand

within each network section, achieved by normalizing the heat demand with respect to its

corresponding reference length. In contrast to that, the network LHD normalizes the heat

demand for the entire network, representing the normalized average heat demand across

the network. The distinction between the local LHD and the network LHD is illustrated

in Figure 4.5. While the network LHD remains constant, the local LHD varies based on

the network length. If different network expansions are considered, the network LHD will

change accordingly unless the local LHD remains constant.

For a specific network expansion 𝑖, the network LHD can be expressed by Equation

(4.12). Here, the counting variable 𝑖represents the section number up to the point of

the network expansion, and the counting variable 𝑖∗represents each segment within this

expansion.

𝑞lin,nw,𝑖 =∑𝑖𝑖∗=1𝑄con,a,𝑖∗,𝑗𝑛rad

Δ𝐿ref 𝑖=𝑋

⏞⏞⏞⏞⏞⏞⏞⏞⏞

∑𝑖𝑖∗=1(𝑞∗

1𝐿bra,𝑖∗+𝑞∗

𝑖(4.12)

Thus, the overall heat demand of all sections is divided by the total reference length of all

sections. The total branch length of this network expansion can be defined as Equation

(4.13), assuming all branch sections within the analytical model share the same length.

𝐿bra,𝑖∗=𝑖∗Δ𝐿bra (4.13)

The term 𝑋in Equation (4.12) can be rewritten using Equation (4.14) by splitting the

sum. Each sum term in Equation (4.14) can be evaluated independently, as shown in

Equations (4.15) and (4.16), leading to Equation (4.17).

𝑋= 𝑖

∑

𝑖∗=1(𝑞∗

1𝐿bra,𝑖∗+𝑞∗

0)= 𝑖

∑

𝑖∗=1𝑞∗

1𝐿bra,𝑖∗+𝑖

∑

𝑖∗=1𝑞∗

0(4.14)

𝑖

∑

𝑖∗=1𝑞∗

0=𝑖𝑞∗

0(4.15)

𝑖

∑

𝑖∗=1𝑞∗

1𝐿bra,𝑖∗=𝑖

∑

𝑖∗=1𝑞∗

1𝑖∗Δ𝐿bra =𝑞∗

1Δ𝐿bra

𝑖

∑

𝑖∗=1𝑖∗=𝑞∗

1Δ𝐿bra 𝑖(𝑖+1)

2(4.16)

𝑋

𝑖=𝑞∗

1Δ𝐿bra 𝑖+1

2+𝑞∗

0(4.17)

100

4.1. Description of the Methodology

The counting variable 𝑖in Equation (4.17) represents the total network expansion divided

by the section length and can be expressed according to Equation (4.18).

𝑖= 𝐿bra,𝑖

Δ𝐿bra

(4.18)

Combining Equations (4.17) and (4.18) leads to Equation (4.19). Substituting Equation

(4.19) into the initial Equation (4.12) leads to Equation (4.20). In this expression, all

sum terms could be dissolved which reduces the number of calculation steps significantly.

Thus, the LHD of the entire network depends just on the coeﬀicients 𝑞∗

0and 𝑞∗

1, the section

length Δ𝐿bra, and the network expansion 𝐿bra,𝑖.

𝑋

𝑖=𝑞∗

2Δ𝐿bra (𝐿bra,𝑖

Δ𝐿bra +1)+𝑞∗

0(4.19)

𝑞lin,nw,𝑖 =𝑞∗

2(𝐿bra,𝑖+Δ𝐿bra)+𝑞∗

0(4.20)

By utilizing the LHD of the entire network, it becomes possible to determine the

annual heat transported through the network and the nominal thermal power entering

the network, as described in Equations (4.21) and (4.22). The annual transported heat

is defined as the LHD of the entire network at the expansion length 𝐿bra,𝑖 multiplied by

the total reference length. Consequently, the nominal thermal power and the annual heat

transported through the linear DHN can be expressed as a continuous function using all

provided input parameters. When considering a SCC, Equations (4.21) and (4.22) can be

directly used to determine the annual transported heat demand and the nominal power

if the network. This allows the determination of pipe diameters, which is explained and

discussed in Section 4.1.4.

𝑄nw,a,𝑖 =𝑞lin,nw,𝑖𝑖Δ𝐿ref =𝑞lin,nw,𝑖𝐿bra,𝑖 Δ𝐿ref

Δ𝐿bra

(4.21)



𝑄nw,n,𝑖 =𝑄nw,a,𝑖

𝜏full,con =𝑞lin,nw,𝑖𝐿bra,𝑖 Δ𝐿ref

𝜏full,con Δ𝐿bra

(4.22)

The determination of the nominal thermal power and the annual transported heat in a

network differs when considering a DCC. This is due to the distributed nature of the pipe

segments in the DCC, resulting in varying nominal power and transported heat at each

segment for different network expansions. The relationship between the segmentation of

the pipe diameters can be observed in Figure 4.6.

SCC:

1 2 34

DCC:

1 2 34

Eﬀective DCC:

1 2 34

Diameter

of SCC

Figure 4.6: Diameter distribution of a SCC, a DCC and an effective DCC for several

network expansions.

101

4. Analytical Model

In the case of a SCC, the pipe diameters increase with network expansions (in the

direction of 𝑖), while remaining constant along the linear distribution path (in the direc-

tion of 𝑖∗). However, when considering a DCC, the pipe diameters change based on both

the network expansion (𝑖direction) and the pipe segmentation (𝑖∗direction). The pipe

diameters increase with expanding the network (𝑖direction) as more consumers are sup-

plied by the linear DHN. Conversely, the pipe diameters of the segments decrease with an

increasing position of 𝑖∗. This decrease is due to consumers extracting a certain volume

flow rate at each pipe segment to meet their heat demand, resulting in lower volume flow

rates at network segments located further away from the heat generator.

To determine an appropriate method for describing a network using a DCC, an effective

DCC configuration is introduced (see the right image of Figure 4.6). This configuration

consists of pipe segments with a consistent diameter throughout each network expansion

𝑖. Consequently, the description of the effective DCC can be compared to a SCC which

enables a simplified cost calculation.

The nominal thermal power of the effective DCC can be described according to Equa-

tion (4.23) using the mean value theorem. In this context, the nominal power of the

network for a given network expansion 𝑖is the averaged thermal power across all pipe

segments 𝑖∗within that expansion.



𝑄nw,n,dcc,eff,𝑖 =1𝑖𝑖

∑

𝑖∗=1 

𝑄nw,n,dcc,𝑖,𝑖∗(4.23)

The thermal power of each pipe segment can be determined using Equation (4.24). Thus,

it can be expressed as the nominal power of the entire network for a particular expansion

𝑖(see Equation (4.22)) minus the nominal power of all preceding2pipe segments. The

power of all preceding pipe segments can be defined in terms of the power of the entire

network, expanded until 𝑖∗−1, as shown in Equation (4.25).



𝑄nw,n,dcc,𝑖,𝑖∗=

𝑄nw,n,𝑖−

𝑄nw,n,𝑖∗−1 (4.24)



𝑄nw,n,𝑖∗−1 =Δ𝐿ref

𝜏full,con Δ𝐿bra 𝑞lin,nw,𝑖∗−1 (𝐿bra,𝑖∗−Δ𝐿bra)(4.25)

To enhance clarity, a substitution based on Equation (4.26) is introduced. By inserting

Equation (4.20) into Equation (4.25), Equation (4.27) is derived.

𝑋= Δ𝐿ref

𝜏full,con Δ𝐿bra

(4.26)



𝑄nw,n,𝑖∗−1 =𝑋[𝑞∗

2(𝐿bra,𝑖∗−Δ𝐿bra +Δ𝐿bra)+𝑞∗

0](𝐿bra,𝑖∗−Δ𝐿bra)(4.27)

A simplification of Equation (4.27) leads to Equation (4.28). By inserting Equation (4.28)

and Equation (4.24) into Equation (4.23), Equation (4.29) is obtained.



𝑄nw,n,𝑖∗−1 =𝑋[𝑞∗

2(𝐿2

bra,𝑖∗−𝐿bra,𝑖∗Δ𝐿bra)+𝑞∗

0(𝐿bra,𝑖∗−Δ𝐿bra)] (4.28)



𝑄nw,n,dcc,eff,𝑖

𝑋=1𝑖𝑖

∑

𝑖∗=1(

𝑄nw,n,𝑖−𝑞∗

2𝐿2

bra,𝑖∗+𝑞∗

2𝐿bra,𝑖∗Δ𝐿bra −𝑞∗

0𝐿bra,𝑖∗+𝑞∗

0Δ𝐿bra)

(4.29)

2The pipe segment closer to the heat producer, located at 𝐿bra,𝑖 =0km.

102

4.1. Description of the Methodology

Equation (4.29) allows the splitting of the sum over all terms into individual sums for each

term. Furthermore, the length of the branch at each position 𝑖∗can be expressed using

Equation (4.13). Solving each sum term individually yields to Equations (4.30) to (4.34).

𝑖

∑

𝑖∗=1 

𝑄nw,n,𝑖 =𝑖 

𝑄nw,n,𝑖 (4.30)

𝑖

∑

𝑖∗=1𝑞∗

2𝐿2

bra,𝑖∗=𝑞∗

2Δ𝐿2

bra

𝑖

∑

𝑖∗=1𝑖∗2 =𝑞∗

2Δ𝐿2

bra 𝑖(𝑖+1)(2𝑖+1)

=𝑞∗

2Δ𝐿2

bra (2𝑖3+3𝑖2+𝑖) (4.31)

𝑖

∑

𝑖∗=1𝑞∗

2𝐿bra,𝑖∗Δ𝐿bra =𝑞∗

2Δ𝐿2

bra

𝑖

∑

𝑖∗=1𝑖∗=𝑞∗

2Δ𝐿2

bra 𝑖(𝑖+1)

2=𝑞∗

4Δ𝐿2

bra 𝑖(𝑖+1) (4.32)

𝑖

∑

𝑖∗=1𝑞∗

0𝐿bra,𝑖∗=𝑞∗

0Δ𝐿bra

𝑖

∑

𝑖∗=1𝑖∗=𝑞∗

0Δ𝐿bra 𝑖(𝑖+1)

2=𝑞∗

2Δ𝐿bra 𝑖(𝑖+1) (4.33)

𝑖

∑

𝑖∗=1𝑞∗

0Δ𝐿bra =𝑖𝑞∗

0Δ𝐿bra (4.34)

By inserting Equations (4.30) into (4.34) into Equation (4.29), Equation (4.35) is derived

which leads to Equation (4.36) after simplification.



𝑄nw,n,dcc,eff,𝑖

𝑋=1𝑖(𝑖

𝑄nw,n,𝑖−𝑞∗

12Δ𝐿2

bra (2𝑖3+3𝑖2+𝑖)+𝑞∗

4Δ𝐿2

bra 𝑖(𝑖+1)

−𝑞∗

2Δ𝐿bra 𝑖(𝑖+1)+𝑖𝑞∗

0Δ𝐿bra)(4.35)



𝑄nw,n,dcc,eff,𝑖

𝑋=𝑞∗

6Δ𝐿2

bra (2𝑖2+3𝑖+1)+𝑞∗

2Δ𝐿(𝑖+1) (4.36)

Substituting 𝑋back into the Equation (4.36) yields to the nominal effective power of the

network for a specific expansion 𝑖, as described by Equation (4.37). The effective annual

transported heat through the network can be determined using Equation (4.38). Dividing

Equation (4.38) by the total reference length Δ𝐿ref,𝑖yields to the effective nominal LHD for

aDCC, as given by Equation (4.39). Here, it can be noted that all sum terms were resolved,

eliminating the need for evaluation at different pipe segment locations when considering

aDCC. This accomplishment represents a significant milestone in the development of an

analytical solution for estimating the heat distribution costs of a DCC.



𝑄nw,n,dcc,eff,𝑖 =Δ𝐿ref

𝜏full,con (𝑞∗

6Δ𝐿bra [2𝐿2

bra,𝑖

Δ𝐿2

bra +3𝐿bra,𝑖

Δ𝐿bra +1]+𝑞∗

2[𝐿bra,𝑖

Δ𝐿bra +1]) (4.37)

𝑄nw,a,dcc,eff,𝑖 =Δ𝐿ref (𝑞∗

6Δ𝐿bra [2𝐿2

bra,𝑖

Δ𝐿2

bra +3𝐿bra,𝑖

Δ𝐿bra +1]+𝑞∗

2[𝐿bra,𝑖

Δ𝐿bra +1]) (4.38)

𝑞lin,nw,dcc,eff,𝑖 =𝑞∗

6(2𝐿bra,𝑖+3Δ𝐿bra +Δ𝐿2

bra

𝐿bra,𝑖)+𝑞∗

2(1+Δ𝐿bra

𝐿bra,𝑖)(4.39)

According to the definitions of the analytical model, three variants of the LHD can

be derived: the local LHD, the network LHD, and the effective network LHD. The local

LHD represents the heat demand at a specific section of the network and reflects the heat

density from the perspective of heat consumers in relation to the length of that section.

103

4. Analytical Model

In contrast, the network LHD describes the average heat demand per total network length

for a particular network expansion. It provides a measure of the average heat consumption

per unit length of the network for a given expansion. On the other hand, the effective

network LHD represents the average heat demand per total network length for a specific

network expansion, taking into account all sections of a DCC. It can be considered a

similar measure to the network LHD, but specifically applicable to the effective DCC

configuration.

It is important to note that the effective network LHD is always smaller than the

network LHD because the heat transported through each network section of the DCC

decreases as the network length increases. This statement holds true unless only a single

pipe segment is considered, in which case the SCC and the DCC scenarios are identical.

How the network LHD and the effective network LHD relates to different shapes of the

local LHD is discussed in detail in Section A.1.7 in the appendix.

4.1.4 Effective Pipe Diameters

The estimation of the pipe capital costs using the analytical model requires information

regarding the pipe diameters of the branch and connection pipes. The pipe diameter can

be determined using Equation (2.45), which involves substituting the pipe friction factor

according to the formulation in Equation (4.2), resulting in Equation (4.40). The volume

flow rate can be expressed by applying the first law of thermodynamics and the law of

continuity.

𝑑in =5

√8

𝑄2

n

𝑉𝑓∗

1𝑒𝑓∗

Δ𝜗2

n𝜋2𝜚𝑐p2Δ𝑝n=5

√

⎷8

𝑄2

n(

𝑄n

𝜚𝑐pΔ𝜗n)𝑓∗

1𝑒𝑓∗

Δ𝜗2

n𝜋2𝜚𝑐p2Δ𝑝n

(4.40)

Simplifying Equation (4.40) further leading to Equations (4.41) and (4.42). In both equa-

tions, the square root term comprises only parameters that are independent of the network

expansion, while the annual heat transported through a pipe depends on the network ex-

pansion. To enhance clarity, the square root term has been substituted with the variable

𝐵∗in Equation (4.42). Consequently, the pipe diameter can be expressed in terms of the

nominal power of a supply and return system. It is important to note that this formulation

is valid only for a continuous description of the pipe diameter.

𝑑in =

𝑄𝑓∗

1+2

√8 𝑒𝑓∗

Δ𝜗2+𝑓∗

n𝜋2𝜚𝑓∗

1+1𝑐p𝑓∗

1+2Δ𝑝n

(4.41)

𝑑in =𝑄𝑓∗

1+2

√8 𝑒𝑓∗

Δ𝜗2+𝑓∗

n𝜋2𝜚𝑓∗

1+1𝑐p𝑓∗

1+2Δ𝑝n𝜏2+𝑓∗

full =𝑄𝑓∗

1+2

a𝐵∗(4.42)

In order to estimate the pipe diameters of the connection pipes, it is necessary to have a

description of the nominal power or the annual heat transported through the connection

pipes. The linear distribution of the local LHD results in a linear distribution of the

nominal power for each connection pipe. To simplify the analysis, an average connection

power is assumed for each network expansion, as given by Equation (4.43). Using this

formulation, the average diameter of the connection pipes for each network expansion can

be expressed as shown in Equation (4.44).

𝑄a,con,𝑖 =𝑄nw,a,𝑖

𝑛rad 𝑖=𝑞lin,nw,𝑖Δ𝐿ref

𝑛rad

(4.43)

𝑑in,con,𝑖 =(𝑞lin,nw,𝑖Δ𝐿ref

𝑛rad )𝑓∗

1+2

5𝐵∗(4.44)

104

4.1. Description of the Methodology

The pipe diameters of the branch depend on the network configuration. In the case of

aSCC, the branch pipe diameter can be directly calculated based on the provided input

parameters. The annual heat transported through the branch of a SCC is similar to the

annual heat demand of the network and can be expressed using Equation (4.45).

𝑄bra,a,scc,𝑖 =𝑄nw,a,𝑖 =𝑞lin,nw,𝑖𝐿bra,𝑖 Δ𝐿ref

Δ𝐿bra

(4.45)

If a simultaneity factor needs to be considered, it can be calculated using Equation (4.46).

The function SF represents the description of the simultaneity factor as given by Equation

(2.7), which depends on the number of consumers.

SF𝑖=SF(𝑛con,𝑖)=SF(𝑛rad 𝐿bra,𝑖

Δ𝐿bra )(4.46)

By combining Equations (4.45) and (4.46), the inner pipe diameter for a specific network

expansion can be determined according to Equation (4.47).

𝑑in,bra,scc,𝑖 =(𝑞lin,nw,𝑖SF𝑖𝐿bra,𝑖Δ𝐿ref

Δ𝐿bra )𝑓∗

1+2

5𝐵∗(4.47)

If a DCC configuration is being considered, a more complex description is needed to

determine the inner pipe diameter of the branch. An iterative evaluation at each pipe seg-

ment of the branch is not feasible in the analytical model as it would significantly increase

computational steps. To address this, an effective diameter for the DCC is utilized. The

effective diameter represents the average continuous pipe diameter along the entire distri-

bution path that would lead to the same distribution costs compared to distributed pipe

diameters, as shown on the right side of Figure 4.6. The effective annual heat transported

through the branch of a DCC can be expressed using Equation (4.48).

𝑄bra,a,dcc,eff,𝑖 =𝑄nw,a,dcc,eff,𝑖 =𝑞lin,nw,dcc,eff,𝑖𝐿bra,𝑖 Δ𝐿ref

Δ𝐿bra

(4.48)

For a given network expansion 𝑖, the effective inner pipe diameter is defined as the sum of

the diameters of all pipe segments of the DCC divided by the total branch length. In the

analytical model, which employs equidistant network segments, the average pipe diameter

can be determined using Equation (4.49). To calculate the inner pipe diameter of each

section 𝑖∗at expansion 𝑖, Equation (4.42) is used, where the annual heat transported

through a pipe segment is defined by Equation (4.24) multiplied by the average consumer

full load hours. The exponent (𝑓∗

1+2)/5is denoted as 𝐾fin the right side of Equation

(4.49) for enhanced clarity. Due to the presence of the exponent 𝐾f, it is not possible

solve the sum term in Equation (4.49) directly. Consequently, using the effective annual

transported heat 𝑄nw,a,dcc,eff,𝑖 to estimate the effective inner pipe diameter will not yield

to the correct value (see Equation (4.50)).

𝑑in,bra,dcc,eff,𝑖 =1𝑖𝑖

∑

𝑖∗=1𝑑in,bra,𝑖,𝑖∗=𝐵∗

𝑖𝑖

∑

𝑖∗=1𝑄𝑓∗

1+2

nw,a,dcc,𝑖,𝑖∗=𝐵∗

𝑖𝑖

∑

𝑖∗=1𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗(4.49)

𝑑in,bra,dcc,eff,𝑖 ≠𝑄𝐾f

nw,a,dcc,eff,𝑖𝐵∗(4.50)

Solving the sum term of Equation (4.49) without using an iterative evaluation is not

feasible using the analytical model at this state. Therefore, an approximation is required

to determine the effective inner pipe diameter of the DCC.

105

4. Analytical Model

For this approximation, the approach given by Equation (4.51) is used. In this equation,

𝜓𝑖represents the ratio of the effective inner pipe diameter of the DCC related to the inner

pipe diameter of the SCC. Thus, if 𝜓𝑖would be known, it could be used to estimate the

effective pipe diameter of the DCC by scaling the known pipe diameter of the SCC with

𝜓𝑖according to Equation (4.52).

𝜓𝑖=𝑑in,bra,dcc,eff,𝑖

𝑑in,bra,scc,𝑖 (4.51)

𝑑in,bra,dcc,eff,𝑖 =𝜓𝑖𝑑in,bra,scc,𝑖 (4.52)

In order to approximate 𝜓𝑖, the effective inner pipe diameter of the DCC is assumed to

be represented by Equation (4.53) introducing the correction factor 𝐾𝜓. By doing so, 𝜓𝑖

can be estimated by relating Equation (4.53) to Equation (4.47) which leads to Equation

(4.54). In this context, the annual transported heat values can be further expressed by

the known values of the effective network LHD 𝑞lin,nw,dcc,eff,𝑖 and the network LHD 𝑞lin,nw,𝑖.

Hence, the only unknown variable at this stage is the correction factor 𝐾𝜓.

𝑑in,bra,dcc,eff,𝑖 ≈𝑄𝐾f

nw,a,dcc,eff,𝑖𝐵∗𝐾𝜓(4.53)

𝜓𝑖≈𝑄𝐾f

nw,a,dcc,eff,𝑖

𝑄𝐾f

nw,a,𝑖 𝐾𝜓=𝑞𝐾f

lin,nw,dcc,eff,𝑖

𝑞𝐾f

lin,nw,𝑖 𝐾𝜓(4.54)

The correction factor 𝐾𝜓can be obtained by transforming Equation (4.53) and substitut-

ing Equation (4.49) which leads to Equation (4.55).

𝐾𝜓=𝑑in,bra,dcc,eff,𝑖

𝐵∗𝑄𝐾f

nw,a,dcc,eff,𝑖 =𝐵∗

𝑖∑𝑖𝑖∗=1𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗

𝐵∗𝑄𝐾f

nw,a,dcc,eff,𝑖 (4.55)

The effective annual transported heat of a DCC 𝑄nw,a,dcc,eff,𝑖 can be expressed using the

mean value theorem which leads to Equation (4.56).

𝑄nw,a,dcc,eff,𝑖 =1𝑖𝑖

∑

𝑖∗=1𝑄nw,a,dcc,𝑖,𝑖∗(4.56)

Inserting Equation (4.56) into Equation (4.55) leads to Equation (4.57). Thus the correc-

tion factor 𝐾𝜓relates the mean value of the annual transported heat of a DCC to the

power of 𝐾fto the effective annual transported heat of the DCC raised to the power of

𝐾f.

𝐾𝜓=1

𝑖∑𝑖𝑖∗=1𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗

𝑖∑𝑖𝑖∗=1𝑄nw,a,dcc,𝑖,𝑖∗)𝐾f=𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗

𝑄𝐾f

nw,a,dcc,eff,𝑖 (4.57)

The effective annual transported heat 𝑄nw,a,dcc,eff,𝑖 can be expressed using Equation (4.38).

Thus, the correction factor 𝐾𝜓depends on the parameters 𝑞∗

0,𝑞∗

1,𝐿bra,𝑖, and Δ𝐿bra. Fur-

thermore, the exponent 𝐾fis dependent on 𝑓∗

1. Therefore, the variables involved in ex-

pressing the correction factor 𝐾𝜓can be determined using Equation (4.58) by utilizing

Equations (4.57), (4.38), and (4.24).

𝐾𝜓=𝑓(𝑞∗

0,𝑞∗

1,𝑓∗

1,𝐿bra,𝑖,Δ𝐿bra)(4.58)

To estimate a numerical value for 𝐾𝜓, Equation (4.57) was solved using a Monte

Carlo simulation3, considering a wide range of input parameters. The specific range of

3This Monte Carlo simulation is not related to the Monte Carlo simulation introduced in Section 3.6.

106

4.1. Description of the Methodology

input parameters utilized is presented in Table 4.3. The coeﬀicient 𝑞∗

0was varied from

𝑞∗

0= 0.25...4.0MWh/m/a to encompass the possible range of the LHD based on the

Monte Carlo simulation performed to analyze the detailed model (refer to Table 3.2) and

maintain a constant distribution of the local LHD. The parameter 𝑞∗

1was varied within the

range of 𝑞∗

1=−1.0...1.0GWh/m2/a to cover a wide range of possible distributions of the

input function for the local LHD. The parameter 𝑓∗

1was varied within a range derived in

Section 4.1.1 to encompass typical ranges for DHN applications. The total branch length

𝐿bra,tot was varied between 0.1...10km, and the number of pipe segments 𝑛seg was varied

between 10...1000. Together, these parameters yield a range of Δ𝐿bra = 0.1...1000m,

covering a wide span of possible values.

Table 4.3: Range of input parameters for the Monte Carlo simulation to estimate the

correction coeﬀicient 𝐾𝜓.

Parameter 𝑞∗

0𝑞∗

1𝑓∗

1𝐿bra,tot 𝑛seg Δ𝐿bra+

Unit [MWh/m/a] [GWh/m2/a] [−] [km] [−] [m]

Minimum 0.25 -1 -0.11 0.1 10 0.1

Maximum 4.0 1 -0.06 10 1000 1000

+Derived value

The Monte Carlo study was conducted with a total of 𝑛=10000simulations. This

resulted in residuals for the mean value of RES𝜇= 4.07⋅10−5 and for the standard

deviation of RES𝜎=9.90⋅10−7, as shown in Equation (3.32). Both residuals are below

the threshold of RES =1.0⋅10−4, indicating that the number of simulations performed is

suﬀicient.

In each simulation of the Monte Carlo study, the actual correction factor 𝐾𝜓was calcu-

lated using Equation (4.57). The resulting density function, representing all Monte Carlo

simulations, is shown in Figure 4.7a. A separation was made between positive and nega-

tive values of 𝑞∗

1. Positive values of 𝑞∗

1indicate an increasing distribution of the local LHD

with respect to the network expansion, while negative values indicate a decreasing distri-

bution. The two probability distributions lead to different outcomes. When considering

an increasing local LHD (𝑞∗

1≥0), a lower variability is observed compared to a decreasing

local LHD (𝑞∗

1<0). The variability can be characterized by the standard deviation 𝜎∗, as

presented in Table 4.4. For 𝑞∗

1≥0, the standard deviation is approximately twice as large

as for 𝑞∗

1<0. Nonetheless, the standard deviation relative to the mean value 𝜇∗is less

than 1.5%, indicating low variability overall. Therefore, the correction factor 𝐾𝜓can be

estimated as constant for all possible scenarios with acceptable accuracy. It is reasonable

to differentiate only between positive and negative values of 𝑞∗

Table 4.4: Evaluation parameters of the Monte Carlo simulation to estimate 𝐾𝜓.

Unit 𝑞∗

1≥0 𝑞∗

1<0 𝑞∗

1≥0&𝑞∗

1<0

𝜇∗(𝐾𝜓)[−] 0.9622 0.9461 0.9542

𝜎∗(𝐾𝜓)[−] 0.0075 0.0138 0.0137

𝜇∗(NRMSE)[%] 0.4870 1.9562 1.2184

𝑚𝑎𝑥(NRMSE)[%] 3.3795 4.8970 4.1645

Subsequently, the error analysis was conducted for the case where 𝐾𝜓is assumed to

be constant. The results are presented in Figure 4.7b, showing the NRMSE between

the actual value of 𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗and its estimated value obtained from Equation (4.59).

107

4. Analytical Model

[-]

(a) 𝐾𝜓

* *

(b) NRMSE

Figure 4.7: Results of the Monte Carlo simulation to estimate the correction coeﬀicient

𝐾𝜓.

The approximation of the correction factor 𝐾𝜓was selected based on the mean value

𝜇∗(𝐾𝜓)provided in Table 4.4. Figure 4.7b illustrates three different scenarios. In the

first scenario (𝑞∗

1≥0), the probability distribution corresponds to cases where 𝑞∗

1≥0.

The average error for this scenario is 𝜇∗(NRMSE)=0.49%, with a maximum error of

𝑚𝑎𝑥(NRMSE) = 3.38%. In the case of negative values only (scenario 𝑞∗

1< 0), the

average error increases to 𝜇(NRMSE)=1.96%, while the maximum error slightly rises

to 𝑚𝑎𝑥(NRMSE)=4.90%. For the scenario without distinction between positive and

negative values (scenario 𝑞∗

1<0&𝑞∗

1>0), the average error is 𝜇∗(NRMSE)=1.22%,

with a maximum error of 𝑚𝑎𝑥(NRMSE)=4.16%.

Consequently, the error can be considered acceptable when estimating the correction

factor 𝐾𝜓as a constant value. It was observed that scenarios with positive values of 𝑞∗

exhibit a lower NRMSE compared to scenarios with negative values of 𝑞∗

1. Thus, it is

recommended to differentiate based on the sign of 𝑞∗

1. Furthermore, the maximum error

encountered during the estimation of 𝐾𝜓remains below five percent, which is considered

acceptable. Consequently, the correction factor 𝐾𝜓will be estimated using Equation (4.60)

for subsequent investigations.

(𝑄𝐾f

nw,a,dcc,𝑖,𝑖∗)approx =𝑄𝐾f

nw,a,dcc,eff,𝑖𝐾𝜓,approx (4.59)

𝐾𝜓,approx ={0.9622if 𝑞∗

1≥0

0.9461if 𝑞∗

1<0 (4.60)

4.1.5 Algorithm to Estimate the LCOH Using the Analytical Model

Utilizing the findings from the previous investigations in this chapter, it becomes possible

to estimate the distribution costs through the employment of the analytical model. This

section outlines the fundamental steps of the algorithm. To utilize the analytical model,

the following input parameters are required: Δ𝐿bra,Δ𝐿con,𝐿bra,𝑖,𝑞∗

0,𝑞∗

1,𝜏full,con,𝑛rad,

𝜗s,n,Δ𝜗n,𝜗amb,𝜁pump,Δ𝑝n,𝑝R,𝜉n,Δ𝑝min,sst,𝑝max,𝜏op,𝜆ins,𝜆soil,𝐿dep,𝐷0,𝐷1,𝐷2,𝑓∗

𝑓∗

1,𝐼0,𝐼1, IR, 𝜏inv,𝑐om,var,dst,𝑐om,fix,dst,𝑐∗el,𝑐th,𝑐inv,pump, and 𝑐inv,sst.

The determination of distribution costs using the analytical model involves the follow-

ing eight steps:

1. Calculation of constants,

2. Estimation of the local, network, and effective LHD,

3. Estimation of the pipe diameters,

108

4.1. Description of the Methodology

4. Calculation of the nominal heat loss load,

5. Calculation of the capital costs,

6. Calculation of the pumping costs,

7. Calculation of the heat loss costs,

8. Estimation of the O&M costs, and

9. Calculation of the overall costs.

The algorithm presented here is utilized to estimate the LCOH for a single branch

length 𝐿bra,𝑖. If distribution costs need to be estimated for multiple network expansions,

the algorithm must be applied for each expansion. Variables that vary with the network

expansion are denoted by the index 𝑖. On the other hand, all other variables remain

constant and are determined only once.

The algorithm begins by calculating the required constants in the first step. In the

second step, values for the local, network, and effective LHD are determined based on the

investigations conducted in Section 4.1.3. Subsequently, the inner pipe diameters for the

connection and branch pipes are calculated using the methodologies introduced and dis-

cussed in Section 4.1.4. Next, the nominal heat losses for both the branch and connection

pipes are computed. Capital costs for the pipes, pumping stations, and substations are

estimated, followed by the computation of pumping costs, heat loss costs, and operation

and maintenance costs. Finally, the overall distribution costs are calculated in the last

step. In the algorithm, variables marked with (∗)are specific to a SCC, while variables

marked with (∗∗)are specific to a DCC. Variables without any markings are required for

both configurations.

1. Calculate constants:

•𝑎=(IR+1)𝜏inv IR

(IR+1)𝜏inv −1

•𝐵∗=5

√8 𝑒𝑓∗

Δ𝜗2+𝑓∗

n𝜋2𝜚𝑓∗

1+1𝑐p𝑓∗

1+2Δ𝑝n𝜏full,con2+𝑓∗

•𝐾f=𝑓∗

1+2

•Δ𝐿ref =Δ𝐿bra +𝑛rad Δ𝐿con

•Δ𝜗m=𝜗s,n−𝜗amb −Δ𝜗n

•𝐿dep,cor =(𝐿dep +𝜆soil

14.6W/m2/K)

2. Calculate the local, network and effective LHD:

•𝑞lin,loc,𝑖 =𝑞∗

1𝐿bra,𝑖+𝑞∗

• In general:

–𝑞lin,nw,𝑖 =𝑞∗

2(𝐿bra,𝑖+Δ𝐿bra)+𝑞∗

–𝑞lin,nw,dcc,eff,𝑖 =𝑞∗

6(2𝐿bra,𝑖+3Δ𝐿bra +Δ𝐿2

bra

𝐿bra,𝑖 )+𝑞∗

2(1+Δ𝐿bra

𝐿bra,𝑖 )(∗∗)

• If Δ𝐿bra <𝐿bra,𝑖/100:

–𝑞lin,nw,𝑖 =𝑞∗

2𝐿bra,𝑖+𝑞∗

–𝑞lin,nw,dcc,eff,𝑖 =𝑞∗

3𝐿bra,𝑖+𝑞∗

2(∗∗)

3. Calculate pipe diameters:

109

4. Analytical Model
•𝑛con,𝑖 =𝑛rad 𝐿bra,𝑖
Δ𝐿bra
• SF𝑖=𝐾1+𝐾2
(1+𝑛con,𝑖
𝐾3)𝐾4(according to Reference [77])
•𝐾1=0.4497,𝐾2=0.5512,𝐾3=53.8438,𝐾4=1.7627
•𝜓𝑖=𝑞𝐾f
lin,nw,dcc,eff,𝑖
𝑞𝐾f
lin,nw,𝑖 𝐾𝜓(∗∗)
•𝐾𝜓={0.9622if 𝑞∗
1≥0
0.9461if 𝑞∗
1<0 (∗∗)
•𝑑in,con,𝑖 =(𝑞lin,nw,𝑖Δ𝐿ref
𝑛rad )𝐾f𝐵∗
•𝑑in,bra,scc,𝑖 =(𝑞lin,nw,𝑖SF𝑖𝐿bra,𝑖Δ𝐿ref
Δ𝐿bra )𝐾f𝐵∗
•𝑑in,bra,dcc,eff,𝑖 =𝜓𝑖𝑑in,bra,scc,𝑖 (∗∗)
4. Calculate heat losses:
• Calculate ratio of outer to inner pipe insulation diameter according to Equation
(4.6) for branch and connection pipes:
–𝜙con,𝑖 =𝐷0
𝑑in,con,𝑖𝐷2+𝐷1
–𝜙bra,scc,𝑖 =𝐷0
𝑑𝐷2
in,bra,scc,𝑖 +𝐷1(∗)
–𝜙bra,dcc,𝑖 =𝐷0
𝑑𝐷2
in,bra,dcc,eff,𝑖 +𝐷1(∗∗)
• Calculate the thermal resistance of the pipes:
–Please note that the index “𝑋” stands either for the connection pipe (index
“con”), the branch pipe of SCC (index “bra,scc”), or the branch pipe of
DCC (index “bra,dcc”).
–𝑅ins,X,𝑖 =1
4𝜋𝐿X,𝑖𝜆ins ln (𝜙X,𝑖)(≈90%of total resistance, see Figure A.3a)
–𝑅soil,X,𝑖 =1
4𝜋𝐿X,𝑖𝜆soil ln (4𝐿dep,cor
𝜙X,𝑖𝑑in,X,𝑖)(≈8%of total resistance, see Figure
A.3a)
–𝑅inter,X,𝑖 =1
4𝜋𝐿X,𝑖𝜆soil ln {[( 2𝐿dep,cor
𝐿bra,𝑖+𝜙X,𝑖𝑑in,X,𝑖)2+1]0.5}(≈2%of total
resistance, see Figure A.3a)
–𝑅X,𝑖 =𝑅ins,X,𝑖+𝑅soil,X,𝑖+𝑅inter,X,𝑖
• Calculate nominal heat loss load:
–
𝑄loss,n,con,𝑖 =Δ𝜗m
𝑅con,𝑖
–
𝑄loss,n,bra,scc,𝑖 =Δ𝜗m
𝑅bra,scc,𝑖 (∗)
–
𝑄loss,n,bra,dcc,𝑖 =Δ𝜗m
𝑅bra,dcc,𝑖 (∗∗)
–
𝑄loss,n,scc,𝑖 =
𝑄loss,n,con,𝑖+
𝑄loss,n,bra,scc,𝑖 (∗)
–
𝑄loss,n,dcc,𝑖 =
𝑄loss,n,con,𝑖+
𝑄loss,n,bra,dcc,𝑖 (∗∗)
5. Calculate the capital costs:
• Capital costs of pipes:
–𝑐cap,pip,con,𝑖 =𝑎Δ𝐿con,𝑖𝑛rad
𝑞lin,nw,𝑖Δ𝐿ref (𝐼0+𝐼1𝑑in,con,𝑖)
–𝑐cap,bra,scc,𝑖 =𝑎Δ𝐿bra
𝑞lin,nw,𝑖Δ𝐿ref (𝐼0+𝐼1𝑑in,bra,scc,𝑖)(∗)
–𝑐cap,bra,dcc,𝑖 =𝑎Δ𝐿bra
𝑞lin,nw,𝑖Δ𝐿ref (𝐼0+𝐼1𝑑in,bra,dcc,eff,𝑖)(∗∗)
110

4.1. Description of the Methodology
• Capital costs of pumps:
–𝑛pump,𝑖 =⌈(𝐿bra,𝑖+Δ𝐿con)Δ𝑝n2(1+𝜉n)+𝑝R+Δ𝑝sst,min
𝑝max ⌉
–𝑐cap,pump,𝑖 =𝑎 𝑐inv,pump 𝑛pump,𝑖
𝜏full,con
–This formulation of the pumps investment costs assume that the pump is
designed according the nominal power of all consumers but not according
to the power of the heat generator. But the error of this simplification is
expected to be small, because the investment costs of the pumps are small
in comparison to the investment costs of the pipes and the heat losses are
comparable small.
• Capital costs of consumer substations:
–𝑐cap,sst =𝑎 𝑐inv,sst
𝜏full,con
•𝑐cap,scc,𝑖 =𝑐cap,pip,con,𝑖+𝑐cap,pip,bra,scc,𝑖+𝑐cap,pump,𝑖+𝑐cap,sst (∗)
•𝑐cap,dcc,𝑖 =𝑐cap,pip,con,𝑖+𝑐cap,pip,bra,dcc,𝑖+𝑐cap,pump,𝑖+𝑐cap,sst (∗∗)
6. Calculate pumping costs:
•𝑐pump,𝑖 =𝑐el 2(𝐿bra,𝑖+Δ𝐿con)Δ𝑝n(1+𝜉n)+Δ𝑝min,sst
Δ𝜗n𝜁pump 𝜚𝑐p
7. Calculate heat loss costs:
•𝑐loss,con,𝑖 =𝑐th 𝜏op 𝑛rad
𝑞lin,nw,𝑖Δ𝐿ref 
𝑄loss,n,con,𝑖
•𝑐loss,bra,scc,𝑖 =𝑐th 𝜏op Δ𝐿bra
𝑞lin,nw,𝑖𝐿bra,𝑖Δ𝐿ref 
𝑄loss,n,bra,scc,𝑖 (∗)
•𝑐loss,bra,dcc,𝑖 =𝑐th 𝜏op Δ𝐿bra
𝑞lin,nw,𝑖𝐿bra,𝑖Δ𝐿ref 
𝑄loss,n,bra,dcc,𝑖 (∗∗)
•𝑐loss,scc,𝑖 =𝑐loss,con,𝑖+𝑐loss,bra,scc,𝑖 (∗)
•𝑐loss,dcc,𝑖 =𝑐loss,con,𝑖+𝑐loss,bra,dcc,𝑖 (∗∗)
8. Calculate O&M costs:
•𝑐om =𝑐om,fix +𝑐om,var
•𝑐om,fix =𝑐om,fix
𝜏full,con =0ct/kWh
•𝑐om,var = 𝑐om,var
9. Calculate the overall distribution costs:
•𝑐dst,scc,𝑖 =𝑐cap,scc,𝑖+𝑐pump,𝑖+𝑐loss,scc,𝑖+𝑐om (∗)
•𝑐dst,scc,𝑖 =𝑐cap,dcc,𝑖+𝑐pump,𝑖+𝑐loss,dcc,𝑖+𝑐om (∗∗)
This given algorithm leads to several differences compared to the detailed model:
1. In the analytical model, the capital costs for pumping stations are calculated based
on consumer power, whereas in the detailed model, they are calculated based on the
heat generator power.
2. The analytical model does not account for individual consumer properties such as
nominal connection power, full load hours, and connection length. Instead, it con-
siders linearly distributed local LHD along the linear distribution path.
3. The pipe friction factor in the analytical model is estimated using a non-linear re-
gression function to avoid iterative calculations.
111

4. Analytical Model

4. The analytical model does not consider non-equidistant lengths of pipe segments in

the branch.

5. The outer and inner diameters of pipe insulation in the analytical model are esti-

mated using regression functions, whereas the detailed model utilizes actual values

from pipe manufacturer data.

6. The pipe diameters in a DCC are calculated for each section in the detailed model,

whereas the analytical model estimates an effective diameter.

7. The variable O&M costs in the analytical model are based on annual consumed heat,

whereas in the detailed model, they are based on annual produced heat.

4.2 Comparing the Characteristic Function to the Detailed

Model

After introducing the analytical model in the previous section, the characteristic functions

obtained from the analytical model are compared to those obtained from the detailed

model. This comparison enables the evaluation of each component of the distribution

costs along the linear distribution path for a given example. To assess the characteristic

functions, three different variants of the default model were examined, each differing in

the distribution of the local LHD. These distributions include constant, falling, and rising

profiles, as illustrated in Figure 4.8. The input parameters were chosen in accordance

with the default parameters of the detailed model, as outlined in Table 3.2. Additionally,

specific input parameters used by the analytical model are presented in Table 4.5. The

approximation factors for the pipe friction factors 𝑓∗

0and 𝑓∗

1were estimated for each

scenario based on the corresponding input parameters. Thus, the actual pipe friction

factor for each case was calculated using Equation (4.4), and the estimation factors 𝑓∗

and 𝑓∗

1were obtained by applying the regression function described in Equation (4.2).

This approach ensures a close fit to the actual pipe friction factor used in the detailed

model.

0 2 4 6 8 10

Branch length Lbra [km]

Local linear heat density

qlin,loc[MWh/m/a]

Constant Falling Rising

Figure 4.8: Linear distribution of local LHD to validate the distribution of the characteris-

tic function of the analytical model against the results obtained from the detailed model.

The investigation results are presented in Table 4.6 as the RMSE (absolute error) and

NRMSE (relative error) for all scenarios. Overall, there is a strong agreement between

the results obtained from the analytical and detailed models. When considering the total

LCOH of the SCC, the highest RMSE value is observed in the rising scenario, with an error

of RMSE =0.13ct/kWh corresponding to NRMSE =1.07%. Conversely, the error in the

total LCOH for the DCC shows only a slight increase. The rising scenario also yields the

highest NRMSE value for the total LCOH in the DCC, which is approximately NRMSE =

1.33%. Therefore, the proposed estimation of distribution costs for a DCC using an

112

4.2. Comparing the Characteristic Function to the Detailed Model

Table 4.5: Specific input parameters of the analytical model to compare the distribution

of the characteristic function obtained from the detailed model.

Parameter Unit Default Value Minimum value Maximum value

𝐷0[−] 0.0282 - -

𝐷1[−] 1.4814 - -

𝐷2[−]1.1777 --

𝑞∗

0(constant) [MWh/m/a] 2.0 0.25 4.0

𝑞∗

1(constant) [GWh/m2/a] 0.0 - -

𝑞∗

0(falling) [MWh/m/a] 4.0 - -

𝑞∗

1(falling) [GWh/m2/a] -0.4 - -

𝑞∗

0(rising) [MWh/m/a] 0.0 - -

𝑞∗

1(rising) [GWh/m2/a] 0.4 - -

effective branch pipe diameter demonstrates excellent performance for the given example.

The capital costs exhibit RMSE values below 0.08ct/kWh (NRMSE = 1.11%) for all

scenarios, indicating very low errors. The heat loss LCOH exhibit relatively larger relative

errors, with the highest error observed in the falling scenario of the DCC (NRMSE =

4.73%). This discrepancy is primarily due to the estimation of inner and outer insulation

diameters in the analytical model and the estimation of the inner pipe diameter using an

effective diameter. No significant error was observed in the pumping LCOH between the

analytical and detailed models, as the calculation method for pumping LCOH remains

consistent in both models.

Table 4.6: Error of the shape of the characteristic function of the analytical model in

comparison of the characteristic function obtained from the detailed model for several

components of the distribution costs.

Scenario Error Unit Total Capital Heat loss Pumping

SCC

Constant RMSE [ct/kWh] 0.0727 0.0244 0.0368 0.0

NRMSE [%] 1.28 0.73 3.36 0.0

Falling RMSE [ct/kWh] 0.0549 0.0247 0.0227 0.0

NRMSE [%] 1.12 0.87 2.78 0.0

Rising RMSE [ct/kWh] 0.1293 0.052 0.0755 0.0

NRMSE [%] 1.07 0.71 2.16 0.0

DCC

Constant RMSE ct/kWh 0.0262 0.0213 0.0237 0.0

NRMSE [%] 0.51 0.74 2.44 0.0

Falling RMSE [ct/kWh] 0.0555 0.0379 0.0331 0.0

NRMSE [%] 1.28 1.59 4.73 0.0

Rising RMSE ct/kWh 0.1529 0.0762 0.0656 0.0

NRMSE [%] 1.33 1.11 1.97 0.0

A graphical comparison of the characteristic functions is provided for the worst-case

scenario in Figure 4.9. This scenario corresponds to the DCC in the rising scenario and

displays the RMSE and NRMSE values for the total LCOH across all considered scenar-

ios. Even in this worst-case scenario, the differences between the characteristic functions

of the detailed and analytical models are barely noticeable. Therefore, the remaining char-

113

4. Analytical Model

acteristic functions are not presented here but can be found in Figures A.16–A.17 in the

appendix.

0 2 4 6 8 10

Branch length Lbra [km]

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

Figure 4.9: Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a rising distribution of the local LHD assuming a

DCC.

4.3 Evaluation Against Monte Carlo Data

In Section 3.6, a Monte Carlo simulation was conducted to examine the correlation between

input parameters and distribution costs. This simulation generated a database of 10000

cases, referred to as the “regular Monte Carlo data” hereafter. This database can be used to

validate the results obtained from the analytical model and assess the prediction accuracy

of the analytical model compared to the detailed model across a wide range of potential

input parameters. However, the regular Monte Carlo data has three limitations when it

comes to validating the analytical model against the detailed model. The first limitation

refers to the local LHD in the regular Monte Carlo data, which only considers a constant

value. As a result, it cannot account for scenarios with rising or falling distributions of

the local LHD. The second limitation is associated with the equidistant network used

to generate the regular Monte Carlo data using the detailed model. This means that

variations in the lengths of branch segments or connection pipes cannot be considered.

However, variations in these lengths are crucial for validation since the analytical model

assumes an equidistant network. The error arising from this assumption in the analytical

model cannot be analyzed using the regular Monte Carlo data. The third limitation of

the regular Monte Carlo data involves the variability of consumer parameters, specifically

full load hours and nominal power. In the data, these parameters are kept constant for

all consumers in the network without any variability. A more realistic assumption would

involve varying full load hours and nominal consumer power within the detailed model

network.

For this reason, a second Monte Carlo data set of the detailed model was generated,

referred to as the “advanced Monte Carlo data”. This data set was designed to address

the limitations of the regular Monte Carlo data. To compensate for these limitations, a

degree of variability of DOV =0.5(as defined in Equation (3.31)), was applied. This

degree of variability allows each variable to randomly fluctuate up to fifty percent around

its mean value.

To address these limitations, the fluctuations were applied to parameters such as the

nominal consumer power 

𝑄n,con,𝑖,𝑗, nominal consumer full load hours 𝜏full,con,𝑖,𝑗, connection

length Δ𝐿con,𝑖,𝑗, and length of each branch segment Δ𝐿bra,𝑖 within the detailed model.

This introduced non-equidistant network structures and variability in consumer full load

hours and nominal power in the advanced Monte Carlo data set. Although these variables

114

4.3. Evaluation Against Monte Carlo Data

were modified within the network, their mean values remained similar to those obtained

from the regular Monte Carlo data set.

Furthermore, it should be noted that the variation in the length of each branch segment

resulted in a varying total number of branch segments in the advanced Monte Carlo data

set. The number of segments ranged from 𝑛lin =90...268in the advanced Monte Carlo

data set, whereas it was 𝑛lin =179in the regular Monte Carlo data set. Additionally, the

nominal consumer power was adjusted to induce a rising or falling trend in the resulting

local LHD. This principle is illustrated in Figure 4.10.

Figure 4.10: Distribution of the nominal consumer power of the detailed model along the

linear distribution path according the regular and advanced Monte Carlo data base.

In the regular Monte Carlo data set, the consumer power for each consumer remains

constant, resulting in a constant local LHD along the linear distribution path. In contrast,

the advanced Monte Carlo data set uses a variable slope, which can be positive or negative,

for the local LHD. The average local LHD in the advanced Monte Carlo data set is similar

to that of the regular Monte Carlo data set at maximum network expansion. Consequently,

the advanced Monte Carlo data set is only suitable for validation purposes under full

network expansion. Comparing scenarios with different network LHD values would be

invalid. Additionally, the slope boundaries were constrained to ensure that the local LHD

never falls below zero for any network expansion.

To obtain the results of the analytical model, the input parameters of all advanced

Monte Carlo cases were adapted to the analytical model. The prediction accuracy of the

analytical model was assessed by comparing the characteristic functions to the results

obtained from the detailed model. For each case and cost component (total, capital,

pumping, and heat loss), the RMSE and NRMSE were calculated. The results of this

analysis are presented in Figure 4.11, which shows histograms of the resulting NRMSE and

RMSE for each distribution cost component and configuration. The NRMSE represents

the relative error in the shape of the characteristic function between the detailed and

analytical models, while the RMSE represents the total error. Table 4.7 provides the

mean, standard deviation, and maximum error values obtained from the analysis.

Based on the results presented in Figure 4.11 and Table 4.7, a strong overall agreement

between the analytical model and the detailed model can be seen. The mean NRMSE

of the total distribution costs for both configurations is NRMSE =1.6%, indicating a

high level of agreement in terms of the relative error. Furthermore, the absolute error

analysis also demonstrates a strong agreement for both configurations, with values below

RMSE =0.1ct/kWh.

Moreover, there is no significant difference between the errors of the SCC and DCC

in general. This implies that the chosen configuration does not impact the prediction

accuracy of the analytical model when compared to the detailed model. The strong agree-

ment observed in the capital costs, which are configuration-independent, suggests that the

115

4. Analytical Model

02468

NRMSE total LCOH [%]

200

400

600

Number of cases [-]

DCC

SCC

0 2 4 6 8

NRMSE capital LCOH [%]

200

400

Number of cases [-]

DCC

SCC

0 2 4 6 8 10 12 14

NRMSE heat loss LCOH [%]

200

400

Number of cases [-]

DCC

SCC

0.05 0.10 0.15 0.20 0.25

NRMSE pumping LCOH [%]

100

150

Number of cases [-]

DCC

SCC

(a) NRMSE

0123

RMSE total LCOH [ct/kWh]

1000

2000

Number of cases [-]

DCC

SCC

0.0 0.5 1.0 1.5 2.0

RMSE capital LCOH [ct/kWh]

500

1000

1500

Number of cases [-]

DCC

SCC

0.0 0.5 1.0 1.5 2.0

RMSE heat loss LCOH [ct/kWh]

1000

2000

3000

Number of cases [-]

DCC

SCC

0.000 0.005 0.010 0.015 0.020

RMSE pumping LCOH [ct/kWh]

250

500

750

1000

Number of cases [-]

DCC

SCC

(b) RMSE

Figure 4.11: Comparison of the results obtained from the detailed and the analytical model

according the advanced Monte Carlo data base. Number of bins per histogram is 100.

estimation of the effective pipe diameter for the DCC, as described in Section 4.1.4, leads

to highly accurate results across a wide range of input parameters. In contrast to the

NRMSE of the capital costs, the NRMSE of the heat loss costs is slightly higher, indi-

cating that the detailed model exhibits slightly less accuracy in predicting heat loss costs

compared to capital costs. This discrepancy is primarily attributed to the approximation

of the inner and outer insulation thickness of the pipes utilized in the analytical model

(see Section 4.1.2). When considering the RMSE, both capital and heat loss costs exhibit

similar values. The error in the pumping LCOH is minimal because the same description

is employed in both the analytical and detailed models. However, the pumping cost error

is not exactly zero since the results of the analytical model were adjusted to match the

branch lengths of the detailed model for comparability. This adjustment involves a linear

interpolation, leading to a small error evident in the results.

When considering the worst-case scenario, a maximum relative error of the total

116

4.4. The Accuracy of the Analytical Model in Context

Table 4.7: Mean and standard deviation of evaluation parameters of comparison between

detailed and analytical model of the advanced Monte Carlo data base.

Error Configuration Parameter Unit Total Capital Heat loss Pumping

NRMSE

SCC 𝜇∗[%] 1.62 1.99 3.76 0.13

𝜎∗[%] 0.71 0.83 1.42 0.06

Maximum [%] 8.61 9.28 13.91 0.26

DCC 𝜇∗[%] 1.60 2.07 4.17 0.13

𝜎∗[%] 0.73 0.79 1.75 0.06

Maximum [%] 9.25 8.69 14.53 0.26

RMSE

SCC 𝜇∗[ct/kWh] 0.1537 0.1147 0.0759 0.0017

𝜎∗[ct/kWh] 0.1600 0.1138 0.1098 0.0019

Maximum [ct/kWh] 3.6957 1.9676 2.2828 0.0205

DCC 𝜇∗[ct/kWh] 0.1393 0.1065 0.0756 0.0017

𝜎∗[ct/kWh] 0.1500 0.1038 0.1053 0.0019

Maximum [ct/kWh] 3.6221 1.8843 2.2470 0.0205

LCOH at NRMSE =9.25%and a corresponding maximum absolute error of RMSE =

3.62ct/kWh were identified. Although this is significantly higher than the average error,

it still falls within an acceptable range, considering that the analytical model is primarily

intended for early planning stages. This worst-case scenario was investigated, character-

ized by a very low LHD of 𝑞lin =0.05MWh/m/a, a very high nominal supply temperature

of 𝜗s,n=118.1°C, and a very high value of the heat conductivity coeﬀicient of the insu-

lation of 𝜆ins =0.055W/m/K. This extreme scenario results in a high relative heat loss

of approximately 44%. Consequently, the thermal power of all consumers is much lower

than that of the heat generator, leading to an underestimation of pump investment costs

by the analytical model. This is because the investment costs of the pumping stations

were derived based on the nominal power of all consumers instead of the nominal power

of the heat generator used in the detailed model.

Therefore, the assessment of the predictive accuracy of the analytical model utilizing

the advanced Monte Carlo data demonstrated excellent results. The average error, which

is less than two percent, exceeds the performance of the regression models discussed in

Sections 3.6.5 and 3.6.6.

4.4 The Accuracy of the Analytical Model in Context

So far, the analytical model has only been assessed in comparison to the results of the

detailed model, as discussed in Sections 4.2 and 4.3. In this section, the prediction accuracy

of the analytical model will not only be compared to the detailed model but also to other

prediction methods introduced and discussed in this thesis, as well as some sub-versions of

these methods. The aim is to provide context for the prediction accuracy of the analytical

method in comparison to other methods. The reference model for this assessment is

the DCC of the advanced Monte Carlo data base, as introduced in Section 4.3. The

DCC configuration is selected as it represents a more realistic approach to real-life DHNs

compared to the SCC. The methods considered for comparison are as follows:

• Single input regression model (Regression-1D)

• Multiple input regression model (Regression-MD)

117

4. Analytical Model

• Coarse estimation using the SCC-version of the analytical model (Analytical-SCC

(coarse))

• Estimation using the results of the SCC-version of the analytical model and subtract-

ing the average offset between the DCC and SCC versions (Analytical-SCC (minus

avg))

• Estimation using the DCC version of the analytical model (Analytical-DCC)

Both regression models can be classified as black-box model approaches, while the

remaining models can be classified as white-box models. Black-box models are charac-

terized by their input and output variables without specific knowledge of the internal

processes. On the other hand, white-box models are characterized by a deterministic sys-

tem description. In this context, the four remaining model approaches can be categorized

as white-box models. The single and multiple input regression models refer to the re-

gression models introduced in Sections 3.6.5 and 3.6.6, respectively. The Analytical-SCC

and Analytical-DCC models correspond to the SCC and DCC versions of the analytical

model, as introduced in Section 4.1 of this chapter. The Analytical-SCC (coarse) and

Analytical-SCC (minus avg) models represent simplified versions of the analytical model,

as described in this section.

Analytical-SCC (coarse) This model represents a rough cost estimation that an en-

gineer could perform to estimate the distribution costs of a linear DHN in a specific area

using common engineering and planning methods. The model assumes a uniform design of

the main branch, similar to the SCC. It neglects the connection pipes and the simultaneity

of multiple consumers. Neglecting the connecting pipes is expected to result in an underes-

timation, while neglecting simultaneity may lead to an overestimation of the distribution

cost. These simplifications may partially compensate each other. Additionally, the model

assumes a constant local LHD and neglects the thermal resistance resulting from the in-

teraction of two pipes in the ground, as well as the correction of pipe depth. This model

is derived by adapting the algorithm of the analytical model (see Section 4.1.5) with the

following parameter settings: 𝑞∗

1=0GWh/m2/a, 𝑅inter =0K/W, and 𝐿dep,cor =𝐿dep.

Analytical-SCC (minus avg) The second additional comparison model involves esti-

mating the distribution costs of the DCC using the results of the SCC from the analytical

model. In Section 3.6.3, the offset between the distribution costs of the DCC and the

SCC was analyzed using the regular Monte Carlo data base. The investigation revealed

an average relative offset of (𝑐dst,dcc/𝑐dst,scc/𝑐dst,dcc)=0.0851. By applying this relative

offset, the distribution costs of the DCC in the analytical model were estimated using

Equation (4.61). This simplification allows obtaining the distribution costs of the DCC

without calculating the effective pipe diameter.

𝑐dst,dcc =𝑐dst,scc(1−0.0851) (4.61)

The results of this final evaluative investigation are presented as relative errors in

Figure 4.12. It is important to note that the results presented in this section are related

to the prediction accuracy of the total distribution costs at a branch length of 𝐿bra =

10km. In contrast, the results presented in Sections 4.2 and 4.3 compare the shapes of

the characteristic functions for various network expansions.

The single input regression model (Regression-1D) exhibits the highest relative error

of NRMSE =56.8%. This error is slightly higher than the error shown in Table 3.9. The

reason for this discrepancy is that the NRMSE of the single input regression model shown

118

4.4. The Accuracy of the Analytical Model in Context

0 10 20 30 40 50 60

NRMSE [%]

Regression-1D

Regression-MD

Analytical-SCC

(coarse)

Analytical-SCC

(minus avg)

Analytical-DCC

56.83%

35.7%

20.81%

11.95%

4.8%

1.29%

Figure 4.12: Prediction error of several cost prediction methods in comparison to the costs

obtained from the detailed model using the advanced Monte Carlo data base.

in Figure 4.12 is compared to the advanced Monte Carlo data base instead of the regular

one. By utilizing the multi-input regression model with all available input parameters, the

relative error can be reduced to NRMSE =35.7%. Consequently, the usage of more input

parameters enhances the accuracy of the regression model by approximately one-third.

Nevertheless, both easily applicable regression models show relatively low accuracy when

compared to the advanced Monte Carlo data base.

The accuracy can be significantly improved by utilizing the coarse white-box estimation

based on common engineering and planning methods. The relative error of the correspond-

ing model (analytical-SCC (coarse)) can be reduced to NRMSE =20.8%, which is more

than a 50%reduction compared to the single input regression model. Hence, employing

a rough estimation using a white-box approach leads to a substantial decrease in relative

prediction error compared to a regression (black-box) approach. However, a relative error

of NRMSE =20%still represents a relatively coarse estimation rather than an accurate

model.

By incorporating a few additional computational steps, the relative prediction accuracy

can be greatly improved by utilizing the analytical model introduced in this chapter (see

analytical-SCC in Figure 4.12). In this case, a relative error of NRMSE =12.0%can be

achieved by considering only the SCC of the analytical model. This represents a nearly

50%reduction in relative prediction error compared to the coarse model (analytical-SCC

(coarse)).

By subtracting the estimated average offset between the distribution costs of the

SCC from the SCC results of the analytical model, the relative error can be reduced

to NRMSE =4.8%(see analytical-SCC (minus avg) in Figure 4.12). This relative error

is less than a quarter of the relative error of the coarse analytical model (analytical-SCC

(coarse)) and less than 1/10th of the relative error achieved by the single input regression

model. Thus, it represents a significant improvement in prediction accuracy.

If the DCC of the analytical model is used to predict the distribution costs, the relative

error can be reduced to NRMSE =1.3%, which is only 6.2%compared to the coarse ana-

lytical model (analytical-SCC (coarse)) and 2.2%compared to the relative error achieved

by the single input regression model.

Based on these investigations, the analytical model developed in this thesis demon-

strates a substantial improvement in prediction accuracy compared to state-of-the-art

black-box and white-box models. Conversely, the additional computational effort required

for the analytical model increases only to a modest and justifiable extent.

119

4. Analytical Model

4.5 Conclusion and Summary

In this chapter, an analytical model has been introduced to predict the distribution costs

of a linear DHN with reasonable effort. The motivation behind developing the analytical

model was to provide planners and decision makers with a simple approach that does not

require the use of a computer program to assess the suitability of an area to be supplied

by DH. The detailed model presented in Chapter 3 is not suitable for this purpose due

to its high computational requirements, which cannot be easily solved without automated

computer programs. The high computational demand is primarily attributed to the need

for discretizing the branch segments and employing additional interpolations and iterations

within the detailed model. Consequently, the detailed model cannot be readily applied to

accurately estimate the suitability of an area for district heating supply during the early

planning stage. In contrast, the regression models introduced in Section 3.6.5 and 3.6.6

are user-friendly and require reasonable effort, but they yield low accuracy. The developed

analytical model, however, is a relatively simple method that achieves a similar level of

accuracy compared to the detailed model.

To accomplish this, a simplified version of a linear DHN model was employed, leading

to a substantial reduction in complexity compared to the detailed model. Additionally,

all iterative and interpolating steps of the detailed model were replaced with continuous

descriptions. Furthermore, the need for discretizing the network branch to compute a

DCC solution was addressed by introducing an effective pipe diameter and a method for

estimating it.

To assess the prediction accuracy of the analytical model, a comparison was made

between the prediction results and those of the detailed model. This comparison involved

comparing the shapes of the characteristic functions obtained from the default model. The

results indicate a very good agreement between these shapes, with a maximum relative

error of NRMSEmax = 1.33%. Furthermore, the results of the analytical model were

compared to the detailed model using a wide range of potential input parameters from a

Monte Carlo database. This assessment revealed a mean relative error of NRMSE =1.6%

for both SCC and DCC. The maximum relative error was estimated to be NRMSEmax =

9.25%, which occurred in the DCC case.

In the final assessment, the prediction results of the analytical model were compared

to state-of-the-art methods, revealing a significant improvement in prediction accuracy.

When compared to a similar state-of-the-art model, a reduction in relative prediction

error was observed, decreasing from NRMSE =20.8%to NRMSE =1.3%. Moreover, the

computational effort required for the analytical model increased slightly.

Therefore, the developed analytical model offers an accurate and simple to use ap-

proach that can assist planners and decision makers in predicting the suitability of an

area for district heating with enhanced prediction accuracy and reasonable effort com-

pared to state-of-the-art methods.

120

5 Conclusions and Outlook

The primary research objective of this thesis was to develop a methodology for estimating

the heat distribution costs of a district heating network without relying solely on past

network design parameters. This approach aimed to integrate the benefits of both the

territorial approach (ease of use) and the detailed network approach (precision). The spe-

cific tasks involved investigating the key parameters that influence heat distribution costs,

developing a method capable of estimating network costs for future 4th and 5th genera-

tion district heating networks, analyzing the accuracy of this method, and determining its

limitations.

To address the research questions, this thesis primarily focused on analyzing the heat

distribution costs of linear networks in detail. However, a method was presented to extend

this analysis to arbitrarily complex district heating networks that do not include loops.

The research showed that the main cost components are capital costs, heat loss costs,

pressure loss costs, and O&M costs. Specifically, this thesis considered these costs in the

form of LCOH. In a linear district heating network, both the capital LCOH and pressure

loss LCOH scale with the network expansion, while heat loss LCOH and O&M LCOH are

largely independent of network expansion.

In contrast to existing territorial network approach methods, the developed methods

(regression and analytical model) achieve higher prediction accuracy by combining informa-

tion related to the current network design parameters and the actual topological structure

of the network. This enables the adaptation of the developed methods to a wide range of

possible DHN design parameters, making them suitable for 4th and 5th generation DHN.

Unlike current territorial assessment approaches, which often rely on a limited set of net-

work design parameters, the developed methods consider a wide range of parameters that

influence heat distribution costs. It has been observed in the literature that, especially in

areas with low heat densities, parameters other than the heat density play a significant

role in estimating heat distribution costs. A Monte Carlo parameter study conducted in

this research confirms this finding. Therefore, the presented methods account for these

factors, making them also suitable for cost prediction in areas with lower heat densities.

Additionally, the network expansion is found to have a significant impact on heat distribu-

tion costs, which is not considered in existing territorial approaches such as the effective

width and linear heat density approach.

Moreover, the developed methods provide specific information about the expected heat

distribution costs under certain conditions. This allows decision-makers and planners to

assess whether the costs fall within an acceptable range based on local conditions such

as taxes, cost substitutes, and consumer price preferences. This advantage is particularly

notable compared to threshold value approaches, where the method results are binary and

do not consider these mentioned conditions.

Unlike detailed network approaches, the developed methods can be utilized by a wide

range of users without requiring extensive tool knowledge. Furthermore, the scope of the

methods is limited to heat distribution costs, as this is the decisive parameter for assessing

the suitability of a DHN within an area. Moreover, the presented methods rely on accurate

data inputs to achieve reliable predictions. Therefore, if the data provided to the methods

is of low quality, the prediction accuracy will also be low.

The presented thesis demonstrates that the expansion of the network significantly

impacts the heat distribution costs. The analysis has shown that a characteristic function

can be determined for each district, which describes the heat distribution costs associated

with network expansion. Generally, longer expansions result in higher heat distribution

121

5. Conclusions and Outlook

costs. In real-life scenarios, consumers often have a maximum price they are willing to

pay. Considering this, the maximum heat price can be utilized to determine the maximum

heat supply radius using the characteristic function. This approach enables the definition

of the size of a single district heating network surrounding a heat generator, up to which

the network can be economically viable, without the need for a detailed planning process.

This can be particularly beneficial during early planning stages as it provides guidance to

planners on how to topologically design district heating networks. Constructing more, but

smaller networks can be advantageous, especially in areas with lower heat densities, as it

allows for a significant reduction in the average pipe diameter.

The investigations conducted in this thesis focused mainly on centralized heat gener-

ation realized by a single plant, which differ from the definition of decentralized feed-in,

a possibility in some 4th and 5th generation DHN types. This limitation applies only to

the DCC, as it considers diameter reductions of pipe segments farther away from the cen-

tralized heat generator. In contrast, the SCC does not account for this limitation, as the

entire network is designed based on the maximum required thermal load at any position.

In reality, the design of a linear DHN with decentralized feed-in would likely involve vary-

ing pipe diameters according to the expected nominal thermal loads of each pipe segment.

Consequently, the pipe investment costs might be lower than those predicted using the

SCC approach. Additionally, the pressure loss costs might not be accurately accounted

for in situations with decentralized feed-in, as pressure losses could be reduced due to

shorter heat transfer fluid transport lengths. Nevertheless, the SCC approach represents

the upper boundary of heat distribution costs even in the context of decentralized feed-in.

In addition to that, the developed methods focused only on district heating networks.

Nevertheless, the analytical method could also be applied to district cooling networks, as it

provides a more general description. However, the presented regression models are limited

to district heating approaches due to the data used, which was exclusively collected for

district heating. If a new database accounting for district cooling was created, a similar

regression method could be derived and applied to district cooling networks.

Moreover, the presented methods assume a DHN configuration consisting of a supply

and a return pipe, which is the most commonly used configuration. However, other network

configurations with multiple network levels are possible. In cases where multiple supply

and return systems are located in parallel, the presented methods could be applied to each

supply and return network individually, with the total network costs being the sum of the

costs of each parallel network. If other network configurations such as three- or multi-

level systems are employed, the presented methods may require additional adaptations.

For networks with multiple levels (more than two) that cannot be described as parallel

supply and return networks, the total network distribution costs could be estimated by

accumulating the heat distribution costs of each network level. To estimate the specific

heat distribution costs of a single network level, a two-level network configuration could

be assumed for the conditions of each level and applied to the developed methods. The

obtained costs could than be divided by two to get the specific costs of a single network

level. However, further investigations are necessary to validate this idea.

The presented methods are limited to linear or radial network topologies, and networks

consisting of loops were excluded from the investigations due to differences in mass flow

distribution compared to linear and radial networks. Further research could be conducted

to find a simplified solution for estimating the costs of network topologies consisting of

loops. One approach could involve identifying areas in the looped network structure where

the mass flow rate is nearly zero at design conditions and theoretically cutting the network

at these positions to reduce complexity and restore a radial network structure. The heat

distribution costs of this modified network structure could then be estimated using the

122

developed methods presented in this thesis. To prevent a diameter reduction using a SCC

may be advantageous in this context. However, additional research is necessary to assess

the feasibility of this approach.

Furthermore, this thesis put a focus on heat distribution costs while assuming knowl-

edge of heat production costs. Further research could be conducted to estimate the heat

production costs for DHS in the context of changing requirement. Although the costs of a

single heat generator can be estimated accurately using existing methods, estimating the

heat generation costs for multiple heat generators in a compound is challenging due to the

unknown optimal size of each heat generator. Thus further research could be conduced in

this field to help engineers to find the optimal size of each heat generator in a generator

in particular in the context of renewable based heat generation.

The presented thesis conducted an investigation that aimed to provide valuable support

for the necessary transformation of the heating sector. The introduced innovative and

simple-to-use methods for estimating the heat distribution costs where developed mainly

to evaluate possible areas being supplied by DHSs at early planning stages, without the

need for an extensive network analysis. This research seeks to inspire and engage a diverse

range of individuals in finding technical suitable and economical feasible solutions for

sustainable heating systems.

123

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132

A Appendix

A.1 Additional Information

A.1.1 Estimation Error Using a Simplified Calculation of Annual

Pumping Energy

In this Section the estimation error is considered if a simplified calculation of the annual

pumping energy is used according to the method suggested by Nussbaumer et al. [68]

introduced in Section 2.4.5.

In this example, a network is selected which consists of one heat generator, a linear

network, and a single consumer located at a distance of 𝐿=1km from the heat generator.

Heat losses were neglected in this example. Refer to Figure A.7 for the exemplary load

profile of the heat consumer. The load profile is characterized by a nominal capacity of



𝑄n=1000kW and full load hours of 𝜏full =2249h/a. The design temperatures of the

DHN were selected to be 𝜗s,n=75°C and 𝜗r,n=60°C. All remaining input parameters

are given in Table A.1.

Table A.1: Nominal parameters used to estimate the electrical pumping costs for several

network control concepts.

Entity Δ𝜗n𝜁pump Δ𝑝nΔ𝑝min,sst 𝜚 𝑐p

Unit [K] [-] [Pa/m] [bar] [kg/m3] [kJ/kg/K]

Value 20 0.75 200 1.0 979.1 4.182

The following control concepts of the DHN were investigated:

1. Constant supply temperature and constant pressure difference control

(T_const_dp_const)

2. Gliding supply temperature and constant pressure difference control

(T_var_dp_const)

3. Constant supply temperature and proportional pressure difference control

(T_const_dp_var)

4. Gliding supply temperature and proportional pressure difference

control (T_var_dp_var)

In the case of constant supply temperature control, the supply temperature of the

network was set to 𝜗s=75°C. For variable temperature control, the supply temperature

was adjusted within the range of 𝜗s=65...75°C based on the ambient temperature. In the

case of constant pressure control, the pump’s differential pressure was set to Δ𝑝pump =

6bar. However, this value was reduced to Δ𝑝pump = 3bar at zero flow rate for the

variable pressure control concept. Please refer to Figures A.1a and A.1b for a graphical

representation of both control concepts.

The simulation and estimation results are presented in Figure A.2. The lowest dif-

ferences between the estimated and simulated electrical power demand, of −61.5%, are

observed when using constant temperature and proportional pressure control. This is due

to the fact that constant temperature control increases the temperature spread, thereby

reducing the required volume flow. At the same time, differential pressure with propor-

tional pressure control in the partial load range is lower than with constant pressure

133

A. Appendix

−20 −10 0 10 20

ϑamb [°C]

ϑs[°C]

Constant temperature control

Gliding temperature control

(a) Temperature

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Vpump [l/s]

∆ppump [bar]

Constant pressure control

Proportional pressure control

(b) Pressure difference

Figure A.1: Temperature and pressure control concepts used to investigate their influence

on the specific pumping costs.

control. Both effects lead to a reduction of the electrical power consumption. In contrast,

the smallest deviations of −10.9%compared to the estimation are observed when using

variable (gliding) temperature and constant pressure control. In general, the simplified

method selected to determine the electrical power demand is a rather conservative esti-

mate. However, since the control concept involved in each case is usually not known in

early planning phases, the chosen estimate represents a conservative but valid method.

Refer to Figures A.1a and A.1b for a graphical illustration of both control concepts.

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

Wel [MWh/a]

Estimation

T const dp const

T var dp const

T const dp var

T var dp var

Figure A.2: Annual power consumption for pumping of the exemplary DHN control con-

cepts.

A.1.2 Assessing the Thermal Resistances of Two Buried Pipes

In this Section, the individual thermal resistances of a two-buried-pipe laying situation is

analyzed and gives additional information related to Section 2.4.6.

To analyze the impact of several thermal resistances, the nominal pipe diameter and

the laying depth were varied, and the corresponding resistances were evaluated. The

chosen input parameters are listed in Table A.2.

The outcomes of the investigation are shown in Figures A.3a and A.3b, illustrating the

relative contributions of the resistances 𝑅1,𝑅2and 𝑅3to the overall resistance 𝑅tot =

𝑅1+𝑅2+𝑅3. Both figures indicate that the resistances 𝑅2and 𝑅3have considerably

less impact in comparison to resistance 𝑅1. The share of 𝑅1with respect to 𝑅tot is

within the range of 89.2...94.8%for nominal pipe diameters of 20...250mm. On the other

hand, the share of 𝑅2ranges between 4.3...8.7% and the share of 𝑅3ranges between

0.9...2.1%, concerning the nominal diameter. In case of a constant nominal pipe diameter,

at increasing laying depths, the share of 𝑅2and 𝑅3in relation to 𝑅tot escalates. Specifically,

134

A.1. Additional Information

Table A.2: Nominal parameters to estimate electrical pumping costs for several network

control concepts. The inner and outer diameters of the pipes insulation (𝑑in and 𝑑out) were

used according to Table A.8 at IC =2.

Entity 𝐿 𝜆ins 𝜆soil 𝛼sur

Unit [m] [W/m/K] [W/m/K] [W/m2/K]

Value 1 0.02 1.0 14.6

at a laying depth of 𝐿dep =3m, the share of 𝑅2and 𝑅3correspond to 9.36%and 3.81%

of 𝑅tot, respectively. To sum up, it can be concluded that the thermal resistance of the

pipe insulation 𝑅1is the principal thermal resistance, accounting for roughly 90%of the

overall resistance. On the contrary, the thermal resistance 𝑅3can be disregarded in most

circumstances as its contribution is less than 3%of the total resistance. The thermal

resistance of the ground 𝑅2is smaller than 𝑅1, however, it is typically non-negligible.

Nominal diameter DN [-]

(a) Pipe diameter 𝐿dep =1.0m.

Laying depth [m]

(b) Laying Depth at DN =100.

Figure A.3: Thermal resistances 𝑅1–𝑅3as function of the nominal pipe diameter and

the laying depth.

In addition to the installation of two parallel pipes in the ground, a single double-

pipe installation is also a frequently utilized method. For a detailed description of this

particular laying configuration, please refer to Section A.1.5 in the appendix.

A.1.3 Inner Pipe Diameter

In this section a detailed derivation of the inner pipe diameter of a single pipe 𝑑in is given.

District heating networks are usually designed by defining a certain specific pressure loss

Δ𝑝n. The pressure loss of a pipe can be formulated by using the Darcy-Weisbach-Equation

given in Equation A.1. Δ𝑝

𝐿=Δ𝑝n=𝑓∗⋅𝜚⋅𝑣2

2⋅𝑑in

(A.1)

The average velocity 𝑣of a pipe can be expressed through Equation A.2 if an incompressible

fluid and a circular shaped pipe is assumed.

𝑣= 4⋅ 

𝑉

𝜋⋅𝑑2

(A.2)

The volume flow rate 

𝑉can be derived from the nominal heat load 

𝑄using Equation A.3.

This equation is a special case of the first law of thermodynamic, in which a steady state

135

A. Appendix

system without a change of kinetic and potential energy is considered.



𝑉 = 

𝑄

𝜚⋅𝑐p⋅Δ𝜗n

(A.3)

Merging Equations A.1, A.2, and A.3 and solving it to 𝑑in leads to Equation A.4.

𝑑in =5

√8⋅ 

𝑄2

n⋅𝑓∗

Δ𝜗2⋅𝜋2⋅𝜚⋅𝑐p2⋅Δ𝑝n

(A.4)

A.1.4 Logarithmic Mean Temperature Difference

The heat transfer in a heat exchanger system is commonly described by the logarith-

mic mean temperature difference, which takes into account the non-constant temperature

change of both heat transferring fluids along the flow path. This is due to the fact that

the rate of heat transfer changes with the temperature difference, and thus the average

temperature difference between the fluids over the heat exchanger length is not simply

the average of the inlet and outlet temperatures. A schematic of this principle for a heat

exchanger in counter flow configuration is shown in Figure A.4a. If the temperature of

one side of the heat exchanger is constant, a temperature distribution as shown in Figure

A.4b develops.

Temperature

Length

(a) Counter flow heat exchanger.

Temperature

Length

(b) Constant temperature on one

side.

Figure A.4: Temperature heating surfaces diagram.

According to Baehr and Stephan [106], the logarithmic mean temperature difference

for a counter flow heat exchanger can be expressed as Equation (A.5), where 𝜗in,1 refers

to the inlet temperature of side 1, 𝜗out,2 refers to the outlet temperature of side 2, 𝜗out,1

refers to the outlet temperature of side 1, and 𝜗in,2 refers to the inlet temperature of side

1. Heat is transferred from side 1 to side 2.

If the temperature of the cooler side of the heat exchanger can be assumed to be

constant, then the logarithmic mean temperature difference can be expressed as Equation

(A.6).

Δ𝜗m=(𝜗in,1−𝜗out,2)−(𝜗out,1−𝜗in,2)

ln(𝜗in,1−𝜗out,2

𝜗out,1−𝜗in,2)(A.5)

Δ𝜗m=(𝜗in,1−𝜗2)−(𝜗out,1−𝜗2)

ln(𝜗in,1−𝜗2

𝜗out,1−𝜗2)(A.6)

136

A.1. Additional Information

A.1.5 Heat Losses of a Single Buried Duo Pipe

Besides an installation of two parallel pipes in the ground a single double pipe installation is

also a common and installation method. See Figure A.5 for a geometrical characterization

of this case. The heat losses can be expressed by Equation (A.7) [82]. The corrected depth

of the pipes 𝐿dep,cor corresponds to Equation 2.29.

Figure A.5: Geometric characterization for heat loss estimations of a single buried duo

pipe based on Reference [82]. Inner and outer diameters refer to the pipe insulation.



𝑄loss =4⋅𝜋⋅𝐿⋅Δ𝜗m⋅𝜆ins

𝑋(A.7)

Where

𝑋=ln(2⋅𝐿pip

𝑑in )+𝐾1⋅ln(𝑑2

out +𝐿2

pip

𝑑2

out −𝐿2

pip )−𝐾2⋅( 𝐿pip

4⋅𝐿dep,cor )2

−(𝑑in

2⋅𝐿pip −𝐾2⋅𝐿pip⋅𝑑in

16⋅𝐿2

dep,cor +2⋅𝐾1⋅𝑑in⋅𝑑2

out⋅𝐿pip

𝑑4

out−𝐿4

pip )2

1−(𝑑in

2⋅𝐿pip )2−𝐾2⋅𝑑in

4⋅𝐿dep,cor +2⋅𝐾1⋅𝑑2

in ⋅𝑑2

out ⋅𝑑4

out−𝐿4

pip

(𝑑4

out−𝐿4

pip)2

(A.8)

𝐾1=𝜆ins −𝜆soil

𝜆ins +𝜆soil

(A.9)

𝐾2=2⋅(1−𝐾2

1−𝐾1⋅( 𝑑out

4⋅𝐿dep,cor )2(A.10)

A.1.6 Price Conversion Rates

Prices are usually influenced by the inflation and change over the years. Thus, prices valid

for different years can usually not directly compared to each other. How average prices

develop over the years in certain country can be expressed by the inflation rate InR, which

describes to the price development during one year of a standardized basket of goods

[107]. Since the inflation rate is usually published for a standardized basket of goods, the

actual inflation rate of a good may vary. Nevertheless, using the average inflation rate

is a straightforward method to convert prices valid for different years. In Table A.3 the

conversion CR rate for different years to the base of 2022 (1st of January) is given for

Germany and the European Union.

A.1.7 Distribution of the local, the network, and the effective network

LHD

According to the definitions of the analytical model, three variants of the LHD can be

derived: the local LHD, the network LHD, and the effective network LHD. How the

137

A. Appendix

Table A.3: Average annual inflation rates and price conversion rates to the base of 2022

(1st of January) for Germany and the EU according to References [108, 107].

Year Germany EU-27

InR [%/a] CR2022 [%] InR [%/a] CR2022 [%]

2010 1.1 116.9 1.8 117.7

2011 2.1 115.8 2.9 115.9

2012 2.0 113.7 2.6 113.0

2013 1.4 111.7 1.3 110.4

2014 1.0 110.3 0.4 109.1

2015 0.5 109.3 0.1 108.7

2016 0.5 108.8 0.2 108.6

2017 1.5 108.3 1.6 108.4

2018 1.8 106.8 1.8 106.8

2019 1.4 105.0 1.4 105.0

2020 0.5 103.6 0.7 103.6

2021 3.1 103.1 2.9 102.9

2022 -100.0 -100.0

network LHD and the effective network LHD relates to the local LHD is discussed in this

Section.

The relationship between the three variants of LHD and the corresponding nominal

power of the network is illustrated in Figure A.6. Three different distributions of the local

LHD are considered, characterized by constant, increasing, and decreasing patterns with

respect to network expansion. The choice of the local LHD distribution ensures that the

network LHD is 𝑞lin,nw =2.0MWh/m/a for a full network expansion of 𝐿bra =10km across

all three distribution types. The coeﬀicients 𝑞∗

0and 𝑞∗

1for these examples are provided in

Table A.4. The length of each branch segment is set to the default value given in Table

3.2 which is Δ𝐿bra =56m.

Table A.4: Coeﬀicients 𝑞∗

0and 𝑞∗

1describing the local LHD according to Equation (4.9) for

the rising, constant and falling scenario.

Unit Constant Rising Falling

𝑞∗

0[MWh/m/a] 2.0 0.0 4.0

𝑞∗

1[GWh/m2/a] 0.0 0.4 -0.4

Considering a constant distribution of the local LHD, the distribution of the network

and effective network LHD, as shown in the left diagram of Figure A.6a, can be observed.

In this case, both the local LHD and the network LHD exhibit a constant value of 𝑞lin,loc =

𝑞lin,nw =2.0MWh/m/a. This occurs because the local LHD remains constant, resulting

in an average value of the same constant value, which is reflected in the network LHD.A

different pattern emerges for the effective LHD. Initially, it matches the values of the local

and network LHD for the length of the first pipe segment, 𝐿bra =0...56m. This is because

the scenario corresponds to a SCC, where only one pipe segment exists. Consequently,

the effective LHD yields to similar results compared to the network LHD. However, as the

network expands, the effective LHD decreases and eventually approaches an asymptote at

𝑞lin,nw,dcc,eff =1.0MWh/m/a, which is half of the network LHD. This behavior arises from

the fact that the effective LHD represents the average heat transported through all pipe

segments for a given network expansion in the DCC. As heat extraction remains constant

138

A.1. Additional Information

0 2 4 6 8 10

Branch length Lbra [km]

Linear heat density

qlin [MWh/m/a]

qlin,loc

qlin,nw

qlin,nw,dcc,eff

0 2 4 6 8 10

Branch length Lbra [km]

Nominal power ˙

Qn[GW]

Qn,nw

Qn,nw,dcc,eff

(a) Constant 𝑞lin,loc

0 2 4 6 8 10

Branch length Lbra [km]

Linear heat density

qlin [MWh/m/a]

qlin,loc

qlin,nw

qlin,nw,dcc,eff

0 2 4 6 8 10

Branch length Lbra [km]

Nominal power ˙

Qn[GW]

Qn,nw

Qn,nw,dcc,eff

(b) Rising 𝑞lin,loc

0 2 4 6 8 10

Branch length Lbra [km]

Linear heat density

qlin [MWh/m/a]

qlin,loc

qlin,nw

qlin,nw,dcc,eff

0 2 4 6 8 10

Branch length Lbra [km]

Nominal power ˙

Qn[GW]

Qn,nw

Qn,nw,dcc,eff

Figure A.6: Comparison of the distribution of the local, the network and the effective

LHD for several scenarios of the local LHD 𝑞lin,loc.

for each pipe segment within a specific network expansion, a continuous reduction of the

thermal power is expected along the linear network path. Averaging the thermal power

of all segments along this path results in half the nominal power compared to the total

nominal power of the entire network.

Associated with the nominal power transported through the network, the distribution

shown in the right diagram of Figure A.6a is observed. A linear increasing trend can be

observed, indicating that as the network expands, the nominal power of the distribution

network also increases. This growth is attributed to the connection of more consumers to

the network as the network expansion progresses. Assuming constant full load hours for

all consumers, the local LHD can be interpreted as a measure of the nominal consumer

power extracted at a specific network position. Consequently, there exists an integral re-

lationship, expressed by Equation (A.11), between the local LHD and the network power.

The effective power transported through the DHN 

𝑄n,nw,dcc,eff is approximately half of the

power transported by the network 

𝑄n,nw. This reduction is associated with the segmenta-

139

A. Appendix

tion of the network when considering a DCC.



𝑄n,nw ∽∫𝑞lin,loc 𝑑𝐿bra (A.11)

When considering a rising distribution of the local LHD, the results shown in Figure

A.6b are obtained. Along the linear distribution path, the local LHD increases from

𝑞lin,loc =0.0MWh/m/a to 𝑞lin,loc =4.0MWh/m/a. The network LHD is half of the local

LHD. This reduction occurs because the network LHD represents the average value of the

local LHD across the entire network length for a given network expansion. In the case

of a rising linear distribution of the local LHD, the network LHD is nearly half of the

local LHD (compare Equations (4.20) and (4.9)). The effective LHD shows a lower trend

compared to the network LHD, which is a consequence of averaging the nominal power at

each pipe segment for a specific network expansion. Analyzing the nominal power of the

network (right side diagram of Figure A.6b), a quadratic relationship with respect to the

network expansion is observed. This behavior is influenced by the integral relationship

between the local LHD and the nominal power of the network as described by Equation

(A.11). A similar trend is observed for the effective nominal power of the network, but

with a lower distribution. This reduction is the outcome of averaging the power at each

pipe segment.

The results obtained for a falling course of the LHD are presented in the left diagram

of Figure A.6c. In this case, the network LHD exhibits a distribution that is higher than

the local LHD. This occurs because the network LHD represents the average value of the

local LHD for a given network expansion. On the other hand, the effective LHD shows

a significantly lower trend compared to the network LHD for network expansions greater

than the length of the first segment. This is because the local LHD is high for low values

of the network expansion, resulting in a substantial amount of heat being extracted at

pipe segments located near the heat generator1. Consequently, only a small amount of

heat is transported to further pipe sections, leading to a significant drop in the effective

network LHD for low network expansion values.

Considering the nominal power for a falling distribution of the local LHD, the diagram

shown on the right-hand side of Figure A.6c is obtained. Here, a steep slope can be

observed for the nominal network power 

𝑄n,nw at low network expansions, while a smaller

slope is observed for higher network expansions. This behavior is directly related to the

local LHD and Equation (A.11). As the network expands further, the local LHD becomes

very low, resulting in minimal increases in the nominal power of the network. Consequently,

the effective nominal power 

𝑄n,nw,dcc,eff even decreases for network expansions greater

than 𝐿bra =7.4km. This is because the effective nominal power represents the average

transported power over all pipe segments for a specific network expansion. When the

network expands further, but the nominal power of these pipe segments remains very low,

the average power over all pipe segments may decrease.

1The heat generator is always located at 𝐿bra =0km

140

A.2. Images

A.2 Images

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

200

400

600

800

1000

Q[kW]

Annual load profile

Sorted annual load profile

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

Month

−20

ϑamb [°C]

Figure A.7: Heat load and temperature profile of an exemplary heat consumer. Heat load

profile has derived using the tool TEASER [109]. The temperature profile of a typical

year was used for climate region 4 (Potsdam) according to Reference [110].

141

A. Appendix

Input

value

min

Input

value

max

Capital

LCOH

min

Capital

LCOH

max

Pump

LCOH

min

Pump

LCOH

max

Q loss

LCOH

min

Q loss

LCOH

max

Total

LCOH

min

Total

LCOH

max

Qcon,n,i,j

τfull,con,i,j

ϑs,n

ϑamb

∆pn

pmax

∆pmin,sst

ξn

ηpump

τop

λins

λsoil

Ldep

∆Lcon,i,j

τinv

c∗

cth

ˆcinv,pump

ˆcinv,sst

-50 % +100 % +40.2 % -23.8 % -0 % +0 % +75 % -47.6 % +36 % -21.9 %

-50 % +100 % +100 % -50 % +0 % +0 % +100 % -50 % +76 % -38 %

-50 % +50 % +28.7 % -11.5 % +223.5 % -50.8 % -50.5 % +25.7 % +58.2 % -14 %

-50 % +50 % +0.1 % -0.1 % +0 % +0 % +8.9 % -8.9 % +1.5 % -1.5 %

-80 % +80 % -5.7 % +13.3 % -79 % +79 % +4.6 % -5.2 % -20 % +24.4 %

-90 % +900 % +0.2 % +3.3 % -0 % -0 % +0.1 % +1.5 % +0.1 % +2.2 %

-50 % +100 % -6.3 % +0 % +0 % +0 % +0 % +0 % -3.8 % +0 %

-62.5 % +56.2 % +37.9 % -12.6 % +0 % +0 % +0 % +0 % +22.8 % -7.6 %

-50 % +100 % -6.3 % +0 % -0.7 % +1.3 % +0 % +0 % -3.9 % +0.3 %

-100 % +100 % -6.3 % +0 % -16.4 % +16.4 % +0 % +0 % -7.4 % +3.6 %

-28.6 % +28.6 % +0 % +0 % +40 % -22.2 % +0 % +0 % +8.8 % -4.9 %

-50 % +0 % +0 % +0 % +0 % +0 % -50 % +0 % -7.9 % +0 %

-66.7 % +100 % -0.6 % +0.9 % +0 % +0 % -65.3 % +89 % -10.7 % +14.6 %

-83.3 % +100 % -0.2 % +0 % +0 % +0 % -21.5 % +2.9 % -3.5 % +0.5 %

-50 % +50 % +0.2 % -0.2 % +0 % +0 % +25.3 % -16.3 % +4.2 % -2.7 %

-50 % +100 % +0 % -0 % +0 % +0 % +0.7 % -0.9 % +0.1 % -0.1 %

-66.7 % +200 % -11.7 % +16.1 % -0.1 % +0.3 % -11.8 % +35.5 % -8.9 % +15.3 %

-50 % +50 % -24.7 % +24.7 % +0 % +0 % +0 % +0 % -14.8 % +14.8 %

-50 % +100 % -22.8 % +53.6 % +0 % +0 % +0 % +0 % -13.7 % +32.2 %

-50 % +33.3 % +55.5 % -12.6 % +0 % +0 % +0 % +0 % +33.4 % -7.6 %

-50 % +100 % +0 % +0 % -50 % +100 % +0 % +0 % -11 % +21.9 %

-50 % +100 % +0 % +0 % +0 % +0 % -50 % +100 % -7.8 % +15.5 %

-50 % +100 % -15.8 % +31.6 % +0 % +0 % +0 % +0 % -9.5 % +19 %

-50 % +100 % -3.4 % +6.8 % +0 % +0 % +0 % +0 % -2 % +4.1 %

<−100%

Extreme

−100% − −40%

High

−40% − −10%

Medium

−2% − −10%

Low

−2% −2%

None

2% −10%

Low

10% −40%

Medium

40% −100%

High

>100%

Extreme

Figure A.8: Results of the sensitivity study using the data obtained from the OFAT

parameter study. The data was derived from evaluation the distribution costs at a network

expansion of 𝐿bra =10km assuming a SCC.

142

A.2. Images

0 20 40 60 80 100

Total LCOH cdst [ct/kWh]

500

1000

1500

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(a) Total LCOH

0 10 20 30 40 50

Capital LCOH ccap,dst [ct/kWh]

500

1000

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(b) Capital LCOH

0 20 40 60

Heat Loss LCOH closs,dst [ct/kWh]

1000

2000

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

0 5 10 15

Pumping LCOH cpump,dst [ct/kWh]

200

400

600

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(d) Pumping LCOH

Figure A.9: Histograms with 100bins of LCOH components. The data was derived from

the Monte Carlo parameter study (𝑛=10000) for a comparable short (𝐿bra =0.5km)

and a fully expanded distribution network (𝐿bra =10km) assuming a SCC.

01234

ln(cdst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(a) Total LCOH

−101234

ln(ccap,dst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(b) Capital LCOH

−2 0 2 4

ln(closs,dst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

−4−2 0 2

ln(cpump,dst/[ct/kWh])

100

200

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(d) Pumping LCOH

Figure A.10: Logarithmic histograms with 100bins of LCOH components of Monte Carlo

parameter study for a comparable short and long distribution network of a SCC.

143

A. Appendix

01234

ln(cdst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(a) Total LCOH

−1 0 1 2 3 4

ln(ccap,dst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(b) Capital LCOH

−2 0 2 4

ln(closs,dst/[ct/kWh])

100

200

300

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

−4−2 0 2

ln(cpump,dst/[ct/kWh])

100

200

Number of cases [-]

Lbra = 10 km

Lbra = 0.5 km

(d) Pumping LCOH

Figure A.11: Logarithmic histograms with 100bins of LCOH components of Monte Carlo

parameter study for a comparable short and long distribution network assuming a DCC.

Qcon,n,i,j

τfull,con,i,j

ϑs,n

∆Lcon,i,j

∆pn

τinv

c∗

cth

pmax

cinv,pump

λins

ϑamb

∆pmin,sst

ξn

ηpump

λsoil

Ldep

cinv,sst

Lbra

qlin

Total

Capital

Heat loss

Pumping

-0.51 -0.44 0.11 -0.11 0.1 0.09 -0.11 0.13 0.09 0.1 -0.07 0.08 0.12 -0.02 0.01 0.02 -0 0.02 -0.04 0.02 0.01 0.03 0.3 -0.59 -0.51 0.18

-0.48 -0.49 0.18 -0.12 0.08 0.03 -0.17 0.21 0.02 -0.01 -0.1 0.11 0 -0 0.02 0.03 -0.01 0.01 -0.01 0 0 0.04 0.25 -0.6 -0.49 0.28

-0.46 -0.29 -0.01 0.16 0.1 -0.04 0 -0.01 0.03 0.26 -0 0.01 0.3 -0.04 0 0 0 -0.01 -0.01 0.04 0.02 0.01 0.06 -0.47 -0.46 -0.01

-0.01 -0.01 0.01 -0.44 0 0.37 -0 0 0.29 -0.01 -0 -0 -0.01 -0 0.01 0.02 -0 0.09 -0.15 0.02 -0.02 0.03 0.47 -0.01 -0.01 0

−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

Figure A.12: Heat map of the Pearson correlation coeﬀicient for several components of

the distribution costs and input parameters. Data was derived from the Monte Carlo

parameter study assuming a SCC.

144

A.2. Images

Figure A.13: Scatter plots including linear and non-linear regression of the relevant input

parameters for correlations coeﬀicients |𝜌|≥0.1. Data was derived from the Monte Carlo

parameter study assuming a SCC.

145

A. Appendix

Figure A.14: Scatter plots and linear regression of the Monte Carlo simulation results for

cases |𝜌|<0.1and consumer heat load, consumer full load hours, interest rate and invest

horizon assuming a DCC.

146

A.2. Images

Figure A.15: Scatter plots and linear regression of the Monte Carlo simulation results for

cases |𝜌|<0.1and consumer heat load, consumer full load hours, interest rate and invest

horizon assuming a SCC.

147

A. Appendix

0 2 4 6 8 10

Branch length Lbra [km]

0.0

2.5

5.0

7.5

10.0

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(a) SCC

0 2 4 6 8 10

Branch length Lbra [km]

0.0

2.5

5.0

7.5

10.0

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(b) DCC

Figure A.16: Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a constant distribution of the local LHD assuming

aDCC.

0 2 4 6 8 10

Branch length Lbra [km]

0.0

2.5

5.0

7.5

10.0

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(a) SCC

0 2 4 6 8 10

Branch length Lbra [km]

0.0

2.5

5.0

7.5

10.0

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(b) DCC

Figure A.17: Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a falling distribution of the local LHD assuming a

DCC.

148

A.2. Images

0 2 4 6 8 10

Branch length Lbra [km]

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(a) SCC

0 2 4 6 8 10

Branch length Lbra [km]

LCOH [ct/kWh]

Total - detailed

Total - analytical

Invest - detailed

Invest - analytical

Pumping - detailed

Pumping - analytical

Heat loss - detailed

Pumping - analytical

(b) DCC

Figure A.18: Comparison of distribution of the characteristic function obtained from the

detailed and the analytical model for a rising distribution of the local LHD assuming a

DCC.

1 2 3 4 5 6 7 8 22

(all)

Number of parameters

RMSE [ct/kWh]

NRMSE [%] and

number of coefficients [-]

Figure A.19: RMSE and NRMSE of the multiple input regression model. The data was

obtained form the Monte Carlo simulation assuming a SCC.

149

A. Appendix

A.3 Tables

Table A.5: Specific investment costs for a standard substation according to minimum

requirements as minimum, maximum and average values [72]. Prices are converted into

Euro using a conversion rate of 1CHF =0.922€[76] as valid for 2020 and converted to

the base year 2022 using conversion rates provided in Table A.3.

Nominal load [kW] Specific capital costs [€/kW]

Minimum Maximum Average

10 373 640 466

25 153 264 194

50 79 151 106

100 45 124 74

200 47 94 59

150

A.3. Tables

Table A.6: Eﬀiciencies and economical parameter of different types of heat generators based on References [71, 8]. Cost data were converted to

the base year 2022 using conversion rates of the European Union provided in Table A.3.

Nominal load 𝑐cap 𝑐om,var 𝑐om,fix 𝜁el4𝜁th4/SPF Lifetime

Type [MW] [€/MW] [€/MWh] [€/MW] [%] [%] [a]

NG HOP 0.5–1020.04–0.26 0.6–2.2 1036–2072 —93–105 25

Wood chip HOP (medium) 5.34–6.8620.62 – 0.84 2.42 – 3.84 28801 – 39057 — 89 – 115 20 – 35

Wood chip HOP (large) 40.4 – 51.820.45 – 0.60 2.41 – 3.81 37089 – 50246 — 89 – 116 20 – 35

Electric HOP 1 – 2520.1 – 0.26 0.52 – 0.93 1036 – 1140 — 98 - 99 20

Waste HOP 36.3 – 38.021.66 – 2.30 7.09 – 9.0 73038 – 100078 — 103 – 109 20 – 35

NG turbine CHP (small) 0.01 – 0.211.24 0.36 N/A 21 – 29 56 – 64 25

NG turbine CHP (medium) 5 – 4010.62 – 1.04 5.18 – 7.25 20202 30 – 38 52 –60 25

NG engine CHP 1 – 1010.93 – 1.14 4.41 – 12.43 7252 – 20720 38 – 46 44 – 52 25

BG engine CHP 1 – 1010.83 – 1.24 6.22 – 13.47 7252 – 20720 36 - 42 48 – 54 25

Wood chip CHP (medium) 2.85 – 2.7915.71 – 7.72 6.68 – 10.78 249676 – 337736 12 – 15 73 – 98 20 – 35

Wood chip CHP (large) 22.2 – 31.813.22 – 4.54 0.98 – 1.55 38643 – 53354 24 – 38 49 – 86 20 – 35

Air source HP (small) 0.3 – 1.520.98 – 1.97 2.25 – 3.28 1036 – 3108 — 275 – 335315 – 40

Air source HP (medium) 1.5 – 5.020.79 – 1.48 2.27 – 3.82 1036 – 3108 — 320 – 370315 – 40

Excess heat HP (small) 0.3 – 1.520.98 – 1.97 1.75 – 2.79 1036 – 3108 — 410 – 450315 – 40

Excess heat HP (large) 1.5 – 5.020.69 – 1.18 2.27 1036 – 3108 — 450 – 490315 – 40

1Related to the electrical power output.

2Related to the thermal power output.

3SPF highly depend on the individual use case, therefore deviations are possible.

4Related to lower heating value.

151

A. Appendix

Table A.7: Pipe and trench cost data according to Reference [70].

DN Trench cost

open field

[Eur/m]

Trench cost

street [Eur/m]

KMR-Uno

[Eur/m]

KMR-Duo

[Eur/m]

MMR-Uno

[Eur/m]

MMR-Duo

[Eur/m]

PMR-Uno

[Eur/m]

PMR-Duo

[Eur/m]

20 93 184 251 179 230 140 134 85

25 93 184 257 183 242 161 145 112

32 93 184 287 207 325 192 190 133

40 93 184 303 220 336 228 205 198

50 120 226 327 233 364 236 303 205

65 120 226 374 271 550 - 349 -

80 139 268 420 303 608 - 427 -

100 156 286 563 412 690 - 459 -

125 175 305 715 528 772 - 490 -

150 184 346 883 659 836 - 531 -

200 203 392 1071 803 - - - -

250 231 439 1521 - - - - -

152

A.3. Tables

Table A.8: Pipe sizing data of KMR pipes according to Reference [70]. Geometric proper-

ties shown in Figures 2.14a (uno) and A.5 (duo).

Nominal Diameter 𝑑pip,in 𝑑ins,in 𝑑pip,in

IC =1 IC =2 IC =3

DN [mm] [mm] [mm] [mm] [mm]

Uno pipe:

20 21.6 26.9 90 110 125

25 28.5 33.7 90 110 125

32 37.2 42.4 110 125 140

40 43.1 48.3 110 125 140

50 54.5 60.3 125 140 160

65 70.3 76.1 140 160 180

80 82.5 88.9 160 180 200

100 107.1 114.3 200 225 250

125 132.5 139.7 225 250 280

150 160.3 168.3 250 280 315

200 210.1 219.1 315 355 400

250 263.0 273.0 400 450 500

Duo pipe:

20 21.7 26.9 125 140 -

25 28.5 33.7 140 160 -

32 37.2 42.4 160 180 -

40 43.1 48.3 160 180 -

50 54.5 60.3 200 225 -

65 70.3 76.1 225 250 -

80 82.5 88.9 250 280 -

100 107.1 114.3 315 355 -

125 132.5 139.7 400 450 -

150 160.3 168.3 450 500 -

200 210.1 219.1 500 630 -

153

A. Appendix

Table A.9: Pipe sizing data of MMR pipes according to Reference [70]. Geometric prop-

erties shown in Figures 2.14a (uno) and A.5 (duo).

Nominal Diameter 𝑑pip,in 𝑑ins,in 𝑑pip,in

IC =1 IC =2 IC =3

DN [mm] [mm] [mm] [mm] [mm]

Uno pipe:

20 22.0 25.5 91 - -

25 30.0 34.0 91 111 -

32 38.9 43.8 111 126 -

40 48.5 54.5 111 126 -

50 60.0 66.5 126 142 -

65 75.8 85.6 178 - -

80 98.0 109.2 178 233 -

100 127.0 142.9 233 - -

125 147.0 162.7 233 - -

150 197.5 218.0 313 - -

Duo pipe:

20 22.0 25.5 111 --

25 30.0 34.0 126 - -

32 38.9 43.8 142 - -

40 48.5 54.5 162 - -

50 60.0 66.5 182 225 -

Table A.10: Pipe sizing data of PMR pipes according to Reference [70]. Geometric prop-

erties shown in Figures 2.14a (uno) and 2.14b (duo).

Nominal Diameter 𝑑pip,in 𝑑ins,in 𝑑pip,in

IC =1 IC =2 IC =3

DN [mm] [mm] [mm] [mm] [mm]

Uno pipe:

20 20.4 25.0 75 90 -

25 26.2 32.0 75 90 -

32 32.6 40.0 90 110 -

40 40.8 50.0 110 125 -

50 51.4 63.0 125 140 -

65 61.4 75.0 140 160 -

80 73.6 90.0 160 180 -

100 90.0 110.0 160 180 -

125 102.2 125.0 180 - -

150 130.8 160.0 250 - -

Duo pipe:

20 20.4 25.0 90 110 -

25 26.2 32.0 110 125 -

32 32.6 40.0 125 140 -

40 40.8 50.0 160 180 -

50 51.4 63.0 180 - -

154

A.3. Tables

Table A.11: Mean, standard deviation, minimum, and maximum value of of several costs

components for two different network expansion. The data was derived from Monte Carlo

parameter study with 𝑛=10000and assuming a SCC.

Entity 𝐿bra 𝜎∗𝜇∗Minimum Maximum

Unit [km] [ct/kWh] [ct/kWh] [ct/kWh] [ct/kWh]

Total LCOH 0.5km 4.49 5.37 0.80 82.13

Total LCOH 10.0km 7.39 11.55 1.49 111.46

Capital LCOH 0.5km 2.78 3.43 0.38 33.18

Capital LCOH 10.0km 4.98 6.86 0.80 52.92

Heat loss LCOH 0.5km 2.15 1.60 0.05 51.71

Heat loss LCOH 10.0km 2.78 2.07 0.06 66.41

Pumping LCOH 0.5km 0.14 0.17 0.01 1.26

Pumping LCOH 10.0km 2.22 2.44 0.07 19.45

Table A.12: Input parameter of the single input regression model according to Equations

3.34 and 3.35 including reached accuracies assuming a SCC.

Model

input 𝑋Type Unit of 𝑋 𝐾1𝐾2[ct/kWh] RMSE [ct/kWh] NRMSE [%]

𝑞lin Lin. [MWh/m/a] −3.4381ct/kWh 14.3821 5.1724 60.42

𝑞lin Non-lin. [MWh/m/a] 0.6546 8.80022 3.7531 43.84

𝑞lin Lin. [kW/m/a] −9.34896ct/kWh 14.8650 5.5000 64.25

𝑞lin Non-lin. [kW/m/a] 0.6545 5.5295 4.9457 57.77

𝐿bra Lin. [km] 0.5889ct/kWh 6.0962 5.4898 71.21

155

A. Appendix

Table A.13: Model coeﬀicients of the multi input regression model for a DCC. See Equation (3.36) for the underlying model equation.

Coeﬀicient Input parameter 𝑋

[𝑋] Unit of coeﬀicient Number of input parameters used in model

1 2 3 4 5 6 7 8 22

K0 1 [ct/kWh] 2.4230e+00 -2.7703e+04 3.3290e+00 -1.5916e+00 -3.0780e+00 -5.1551e+00 -7.0031e+00 -4.8693e+00 -4.2361e+04

K1 𝑞lin/(MWh/m/a)[ct/kWh] 5.3184e+00 1.0031e+01 9.2129e+00 9.2167e+00 9.3080e+00 9.3555e+00 9.4199e+00 9.4286e+00 9.9686e+00

K2 𝑞lin/(MWh/m/a)[−] 8.4402e-01 6.6188e-01 7.0950e-01 7.1255e-01 7.0800e-01 7.0624e-01 7.0465e-01 7.0384e-01 6.6850e-01

K3 𝑞lin/(kW/m)[ct/kWh] - 2.7703e+04 -5.9548e+00 -5.7131e+00 -5.9372e+00 -5.9165e+00 -5.8459e+00 -5.9096e+00 4.2354e+04

K4 𝑞lin/(kW/m)[−] - -9.4820e-05 3.2833e-01 3.4405e-01 3.3630e-01 3.3872e-01 3.4401e-01 3.4115e-01 -6.6275e-05

K5 𝐿bra/m [ct/kWh] - - 5.7775e-04 5.8307e-04 5.8191e-04 5.8073e-04 5.7877e-04 5.7901e-04 5.8208e-04

K6 𝑎/(1/a)[ct/kWh] - - - 6.5796e+01 6.5843e+01 6.6007e+01 6.5893e+01 6.5669e+01 6.5375e+01

K7 𝜆ins/(W/m/K)[ct/kWh] - - - - 4.7031e+01 4.8085e+01 4.7858e+01 4.7886e+01 4.7386e+01

K8 𝐼1/(€/m2)ct/kWh - - - - - 3.3023e-04 3.3158e-04 3.2922e-04 3.2797e-04

K9 𝑐th/(ct/kWh)[ct/kWh] - - - - - - 1.7471e-01 1.7806e-01 1.8056e-01

K10 𝜗s,n/°C+273.15 [ct/kWh] - - - - - - - -2.5941e-02 -2.6155e-02

K11 𝐿con,𝑖,𝑗/m [ct/kWh] - - - - - - - - 3.4630e-02

K12 Δ𝑝n/(Pa/m)[ct/kWh] - - - - - - - - 5.6728e-03

K13 𝑐inv,pump/(€/kW)[ct/kWh] - - - - - - - - 1.0700e-02

K14 𝑐∗el/(ct/kWh)[ct/kWh] - - - - - - - - 5.2074e-02

K15 𝑝max/Pa [ct/kWh] - - - - - - - - -7.4006e-07

K16 𝜗amb/°C+273.15 [ct/kWh] - - - - - - - - -2.6462e-02

K17 𝜂pump [ct/kWh] - - - - - - - - -1.7721e+00

K18 𝜉n[ct/kWh] - - - - - - - - 1.9564e+00

K19 𝜆soil/(W/m/K)[ct/kWh] - - - - - - - - 7.8881e-02

K20 𝑐inv,sst/(€/kW)[ct/kWh] - - - - - - - - 2.8780e-03

K21 𝑝R/Pa [ct/kWh] - - - - - - - - 4.6085e-07

K22 𝑟/m [ct/kWh] - - - - - - - - 1.1604e+03

K23 𝐿dep/m [ct/kWh] - - - - - - - - -2.4359e-02

K24 Δ𝑝min,sst/Pa [ct/kWh] - - - - - - - - 1.3178e-06

RMSE 3.3912 3.2866 2.8215 2.6131 2.5235 2.4582 2.3823 2.3065 2.0205

NRMSE 43.45 42.11 36.15 33.48 32.33 31.49 30.52 29.55 25.89

156

A.3. Tables

Table A.14: Model coeﬀicients of the multi input regression model for a SCC. See Equation (3.36) for the underlying model equation.

Coeﬀicient Input parameter 𝑋

[𝑋] Unit of coeﬀicient Number of input parameters used in model

1 2 3 4 5 6 7 8 22

K0 1 [ct/kWh] 2.4425e+00 -2.9883e+04 2.5713e+00 -2.8970e+00 -4.5541e+00 -7.2316e+00 -9.2504e+00 -6.8636e+00 -3.7194e+04

K1 𝑞lin/(MWh/m/a)[ct/kWh] 6.0670e+00 1.1378e+01 1.0363e+01 1.0367e+01 1.0467e+01 1.0528e+01 1.0599e+01 1.0609e+01 1.1335e+01

K2 𝑞lin/(MWh/m/a)[−] 8.3321e-01 6.5482e-01 7.0736e-01 7.1042e-01 7.0600e-01 7.0400e-01 7.0250e-01 7.0167e-01 6.5996e-01

K3 𝑞lin/(kW/m)[ct/kWh] - 2.9882e+04 -5.7198e+00 -5.5301e+00 -5.7383e+00 -5.7226e+00 -5.6655e+00 -5.7247e+00 3.7186e+04

K4 𝑞lin/(kW/m)[−] - -9.8158e-05 3.6790e-01 3.8285e-01 3.7484e-01 3.7750e-01 3.8262e-01 3.7965e-01 -8.3743e-05

K5 𝐿bra/m [ct/kWh] - - 6.4180e-04 6.4779e-04 6.4654e-04 6.4501e-04 6.4284e-04 6.4312e-04 6.4611e-04

K6 𝑎/(1/a)[ct/kWh] - - - 7.4230e+01 7.4282e+01 7.4494e+01 7.4367e+01 7.4116e+01 7.3750e+01

K7 𝜆ins/(W/m/K)[ct/kWh] - - - - 5.1276e+01 5.2639e+01 5.2389e+01 5.2421e+01 5.1956e+01

K8 𝐼1/(€/m2)[ct/kWh] - - - - - 4.2750e-04 4.2899e-04 4.2633e-04 4.2504e-04

K9 𝑐th/(ct/kWh)[ct/kWh] - - - - - - 1.9290e-01 1.9667e-01 1.9873e-01

K10 𝜗s,n/°C+273.15 [ct/kWh] - - - - - - - -2.9173e-02 -2.9384e-02

K11 𝐿con,𝑖,𝑗/m [ct/kWh] - - - - - - - - 3.0800e-02

K12 Δ𝑝n/(Pa/m)[ct/kWh] - - - - - - - - 5.0650e-03

K13 𝑐inv,pump/(€/kW)[ct/kWh] - - - - - - - - 1.0781e-02

K14 𝑐∗el/(ct/kWh)[ct/kWh] - - - - - - - - 5.2512e-02

K15 𝑝max/Pa [ct/kWh] - - - - - - - - -7.3969e-07

K16 𝜗amb/°C+273.15 [ct/kWh] - - - - - - - - -2.8920e-02

K17 𝜂pump [ct/kWh] - - - - - - - - -1.7558e+00

K18 𝜉n[ct/kWh] - - - - - - - - 1.9905e+00

K19 𝜆soil/(W/m/K)[ct/kWh] - - - - - - - - 9.2093e-02

K20 𝑐inv,sst/(€/kW)[ct/kWh] - - - - - - - - 2.7729e-03

K21 𝑝R/Pa [ct/kWh] - - - - - - - - 5.1027e-07

K22 𝑟/m [ct/kWh] - - - - - - - - 1.5201e+03

K23 𝐿dep/m [ct/kWh] - - - - - - - - -1.8161e-02

K24 Δ𝑝min,sst/Pa [ct/kWh] - - - - - - - - 1.4310e-06

RMSE 3.7206 3.6013 3.0744 2.8303 2.7319 2.6303 2.5437 2.4537 2.2165

NRMSE 43.46 42.07 35.91 33.06 31.91 30.72 29.71 28.66 25.89

157

A. Appendix

Table A.15: Permutation table of pipe friction factor coeﬀicients according to the nominal

pressure loss, the pipes roughness and the nominal temperature related to Equation (4.2).

Δ𝑝n[Pa/m] 𝑟[mm] 𝜗n[°C] 𝑓∗

0[-] 𝑓∗

1[-]

100 0.001 40 -4.599861 -0.100406

100 0.001 80 -4.691732 -0.095794

100 0.001 120 -4.752367 -0.093250

100 0.010 40 -4.619629 -0.103466

100 0.010 80 -4.707192 -0.094654

100 0.010 120 -4.744480 -0.084406

100 0.100 40 -4.371881 -0.072298

100 0.100 80 -4.358229 -0.073867

100 0.100 120 -4.351444 -0.076119

300 0.001 40 -4.634901 -0.098779

300 0.001 80 -4.726897 -0.094708

300 0.001 120 -4.788471 -0.092630

300 0.010 40 -4.653346 -0.099431

300 0.010 80 -4.714345 -0.084843

300 0.010 120 -4.729364 -0.073616

300 0.100 40 -4.320113 -0.074336

300 0.100 80 -4.309860 -0.077769

300 0.100 120 -4.305135 -0.079729

500 0.001 40 -4.651316 -0.098118

500 0.001 80 -4.743701 -0.094346

500 0.001 120 -4.805908 -0.092474

500 0.010 40 -4.663606 -0.096051

500 0.010 80 -4.709016 -0.079570

500 0.010 120 -4.715660 -0.069617

500 0.100 40 -4.296387 -0.076084

500 0.100 80 -4.287805 -0.079571

500 0.100 120 -4.283823 -0.081261

158